Properties

Degree 68
Conductor $ 2^{68} \cdot 3^{34} \cdot 67^{34} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 34·3-s + 4-s − 4·7-s + 2·8-s + 595·9-s − 34·12-s + 16-s + 136·21-s − 68·24-s + 68·25-s − 7.14e3·27-s − 4·28-s − 16·29-s + 4·31-s + 2·32-s + 595·36-s + 12·37-s + 4·43-s − 34·48-s − 88·49-s − 8·56-s − 2.38e3·63-s + 7·64-s + 18·67-s + 1.19e3·72-s + 12·73-s − 2.31e3·75-s + ⋯
L(s)  = 1  − 19.6·3-s + 1/2·4-s − 1.51·7-s + 0.707·8-s + 198.·9-s − 9.81·12-s + 1/4·16-s + 29.6·21-s − 13.8·24-s + 68/5·25-s − 1.37e3·27-s − 0.755·28-s − 2.97·29-s + 0.718·31-s + 0.353·32-s + 99.1·36-s + 1.97·37-s + 0.609·43-s − 4.90·48-s − 12.5·49-s − 1.06·56-s − 299.·63-s + 7/8·64-s + 2.19·67-s + 140.·72-s + 1.40·73-s − 266.·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{68} \cdot 3^{34} \cdot 67^{34}\right)^{s/2} \, \Gamma_{\C}(s)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{68} \cdot 3^{34} \cdot 67^{34}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(68\)
\( N \)  =  \(2^{68} \cdot 3^{34} \cdot 67^{34}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{804} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(68,\ 2^{68} \cdot 3^{34} \cdot 67^{34} ,\ ( \ : [1/2]^{34} ),\ 1 )$
$L(1)$  $\approx$  $0.00828701$
$L(\frac12)$  $\approx$  $0.00828701$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 68. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 67.
$p$$F_p(T)$
bad2 \( 1 - T^{2} - p T^{3} + p T^{5} - p T^{6} + p T^{7} + 3 T^{8} + 7 p T^{9} - T^{10} - 7 p^{3} T^{11} + 3 p^{4} T^{12} + 9 p^{3} T^{13} - p^{3} T^{14} - p^{6} T^{15} - 19 p^{3} T^{16} + p^{8} T^{17} - 19 p^{4} T^{18} - p^{8} T^{19} - p^{6} T^{20} + 9 p^{7} T^{21} + 3 p^{9} T^{22} - 7 p^{9} T^{23} - p^{7} T^{24} + 7 p^{9} T^{25} + 3 p^{9} T^{26} + p^{11} T^{27} - p^{12} T^{28} + p^{13} T^{29} - p^{15} T^{31} - p^{15} T^{32} + p^{17} T^{34} \)
3 \( ( 1 + T )^{34} \)
67 \( 1 - 18 T + 263 T^{2} - 2536 T^{3} + 15916 T^{4} + 21888 T^{5} - 1884372 T^{6} + 29838792 T^{7} - 281899248 T^{8} + 1676593384 T^{9} - 20990344 p T^{10} - 110944885160 T^{11} + 1642510138316 T^{12} - 13488669025792 T^{13} + 63391164577644 T^{14} + 2352888210968 p T^{15} - 6009516040445238 T^{16} + 66140757757941972 T^{17} - 6009516040445238 p T^{18} + 2352888210968 p^{3} T^{19} + 63391164577644 p^{3} T^{20} - 13488669025792 p^{4} T^{21} + 1642510138316 p^{5} T^{22} - 110944885160 p^{6} T^{23} - 20990344 p^{8} T^{24} + 1676593384 p^{8} T^{25} - 281899248 p^{9} T^{26} + 29838792 p^{10} T^{27} - 1884372 p^{11} T^{28} + 21888 p^{12} T^{29} + 15916 p^{13} T^{30} - 2536 p^{14} T^{31} + 263 p^{15} T^{32} - 18 p^{16} T^{33} + p^{17} T^{34} \)
good5 \( 1 - 68 T^{2} + 2334 T^{4} - 53816 T^{6} + 935973 T^{8} - 13067332 T^{10} + 152110281 T^{12} - 302600004 p T^{14} + 13067728266 T^{16} - 98978994384 T^{18} + 660564070641 T^{20} - 3882218964872 T^{22} + 3990358248806 p T^{24} - 88176845698324 T^{26} + 64800202897848 p T^{28} - 925781797724652 T^{30} + 1839080435936258 T^{32} - 3866465311500328 T^{34} + 1839080435936258 p^{2} T^{36} - 925781797724652 p^{4} T^{38} + 64800202897848 p^{7} T^{40} - 88176845698324 p^{8} T^{42} + 3990358248806 p^{11} T^{44} - 3882218964872 p^{12} T^{46} + 660564070641 p^{14} T^{48} - 98978994384 p^{16} T^{50} + 13067728266 p^{18} T^{52} - 302600004 p^{21} T^{54} + 152110281 p^{22} T^{56} - 13067332 p^{24} T^{58} + 935973 p^{26} T^{60} - 53816 p^{28} T^{62} + 2334 p^{30} T^{64} - 68 p^{32} T^{66} + p^{34} T^{68} \)
7 \( ( 1 + 2 T + 50 T^{2} + 104 T^{3} + 1219 T^{4} + 2696 T^{5} + 2827 p T^{6} + 960 p^{2} T^{7} + 248162 T^{8} + 622978 T^{9} + 2631571 T^{10} + 6666872 T^{11} + 24800242 T^{12} + 8606460 p T^{13} + 4318624 p^{2} T^{14} + 479245040 T^{15} + 1637496394 T^{16} + 3482409216 T^{17} + 1637496394 p T^{18} + 479245040 p^{2} T^{19} + 4318624 p^{5} T^{20} + 8606460 p^{5} T^{21} + 24800242 p^{5} T^{22} + 6666872 p^{6} T^{23} + 2631571 p^{7} T^{24} + 622978 p^{8} T^{25} + 248162 p^{9} T^{26} + 960 p^{12} T^{27} + 2827 p^{12} T^{28} + 2696 p^{12} T^{29} + 1219 p^{13} T^{30} + 104 p^{14} T^{31} + 50 p^{15} T^{32} + 2 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
11 \( ( 1 + 75 T^{2} - 68 T^{3} + 2884 T^{4} - 5024 T^{5} + 77848 T^{6} - 17244 p T^{7} + 1676376 T^{8} - 4910496 T^{9} + 30512488 T^{10} - 98245228 T^{11} + 481443968 T^{12} - 1613456096 T^{13} + 6673382908 T^{14} - 22402600284 T^{15} + 82108547374 T^{16} - 266083758016 T^{17} + 82108547374 p T^{18} - 22402600284 p^{2} T^{19} + 6673382908 p^{3} T^{20} - 1613456096 p^{4} T^{21} + 481443968 p^{5} T^{22} - 98245228 p^{6} T^{23} + 30512488 p^{7} T^{24} - 4910496 p^{8} T^{25} + 1676376 p^{9} T^{26} - 17244 p^{11} T^{27} + 77848 p^{11} T^{28} - 5024 p^{12} T^{29} + 2884 p^{13} T^{30} - 68 p^{14} T^{31} + 75 p^{15} T^{32} + p^{17} T^{34} )^{2} \)
13 \( 1 - 202 T^{2} + 20513 T^{4} - 1397384 T^{6} + 71905920 T^{8} - 2984288512 T^{10} + 104162435968 T^{12} - 3147802485304 T^{14} + 6472168930596 p T^{16} - 2021525422384504 T^{18} + 44203317017326148 T^{20} - 888132807415483304 T^{22} + 16517338911140513536 T^{24} - \)\(28\!\cdots\!68\)\( T^{26} + \)\(46\!\cdots\!20\)\( T^{28} - \)\(70\!\cdots\!40\)\( T^{30} + \)\(10\!\cdots\!74\)\( T^{32} - \)\(13\!\cdots\!00\)\( T^{34} + \)\(10\!\cdots\!74\)\( p^{2} T^{36} - \)\(70\!\cdots\!40\)\( p^{4} T^{38} + \)\(46\!\cdots\!20\)\( p^{6} T^{40} - \)\(28\!\cdots\!68\)\( p^{8} T^{42} + 16517338911140513536 p^{10} T^{44} - 888132807415483304 p^{12} T^{46} + 44203317017326148 p^{14} T^{48} - 2021525422384504 p^{16} T^{50} + 6472168930596 p^{19} T^{52} - 3147802485304 p^{20} T^{54} + 104162435968 p^{22} T^{56} - 2984288512 p^{24} T^{58} + 71905920 p^{26} T^{60} - 1397384 p^{28} T^{62} + 20513 p^{30} T^{64} - 202 p^{32} T^{66} + p^{34} T^{68} \)
17 \( ( 1 + 154 T^{2} - 10 T^{3} + 12007 T^{4} - 2236 T^{5} + 633281 T^{6} - 199606 T^{7} + 25348959 T^{8} - 10732600 T^{9} + 817172997 T^{10} - 408252386 T^{11} + 21953994331 T^{12} - 11874224100 T^{13} + 501614298749 T^{14} - 274906109790 T^{15} + 9860145355985 T^{16} - 5169284406096 T^{17} + 9860145355985 p T^{18} - 274906109790 p^{2} T^{19} + 501614298749 p^{3} T^{20} - 11874224100 p^{4} T^{21} + 21953994331 p^{5} T^{22} - 408252386 p^{6} T^{23} + 817172997 p^{7} T^{24} - 10732600 p^{8} T^{25} + 25348959 p^{9} T^{26} - 199606 p^{10} T^{27} + 633281 p^{11} T^{28} - 2236 p^{12} T^{29} + 12007 p^{13} T^{30} - 10 p^{14} T^{31} + 154 p^{15} T^{32} + p^{17} T^{34} )^{2} \)
19 \( 1 - 324 T^{2} + 53806 T^{4} - 6082390 T^{6} + 524625555 T^{8} - 36706106300 T^{10} + 2163415764445 T^{12} - 110167923141938 T^{14} + 4934906438372087 T^{16} - 197035530538406632 T^{18} + 7082329410926741589 T^{20} - \)\(23\!\cdots\!26\)\( T^{22} + \)\(68\!\cdots\!71\)\( T^{24} - \)\(18\!\cdots\!28\)\( T^{26} + \)\(47\!\cdots\!01\)\( T^{28} - \)\(57\!\cdots\!34\)\( p T^{30} + \)\(23\!\cdots\!21\)\( T^{32} - \)\(45\!\cdots\!52\)\( T^{34} + \)\(23\!\cdots\!21\)\( p^{2} T^{36} - \)\(57\!\cdots\!34\)\( p^{5} T^{38} + \)\(47\!\cdots\!01\)\( p^{6} T^{40} - \)\(18\!\cdots\!28\)\( p^{8} T^{42} + \)\(68\!\cdots\!71\)\( p^{10} T^{44} - \)\(23\!\cdots\!26\)\( p^{12} T^{46} + 7082329410926741589 p^{14} T^{48} - 197035530538406632 p^{16} T^{50} + 4934906438372087 p^{18} T^{52} - 110167923141938 p^{20} T^{54} + 2163415764445 p^{22} T^{56} - 36706106300 p^{24} T^{58} + 524625555 p^{26} T^{60} - 6082390 p^{28} T^{62} + 53806 p^{30} T^{64} - 324 p^{32} T^{66} + p^{34} T^{68} \)
23 \( 1 - 390 T^{2} + 77399 T^{4} - 452078 p T^{6} + 1061535793 T^{8} - 87680564496 T^{10} + 6092118022573 T^{12} - 365572082566210 T^{14} + 19305682322219015 T^{16} - 909807503167401174 T^{18} + 38667741337071246813 T^{20} - \)\(14\!\cdots\!96\)\( T^{22} + \)\(52\!\cdots\!10\)\( T^{24} - \)\(17\!\cdots\!68\)\( T^{26} + \)\(51\!\cdots\!70\)\( T^{28} - \)\(14\!\cdots\!44\)\( T^{30} + \)\(36\!\cdots\!18\)\( T^{32} - \)\(87\!\cdots\!56\)\( T^{34} + \)\(36\!\cdots\!18\)\( p^{2} T^{36} - \)\(14\!\cdots\!44\)\( p^{4} T^{38} + \)\(51\!\cdots\!70\)\( p^{6} T^{40} - \)\(17\!\cdots\!68\)\( p^{8} T^{42} + \)\(52\!\cdots\!10\)\( p^{10} T^{44} - \)\(14\!\cdots\!96\)\( p^{12} T^{46} + 38667741337071246813 p^{14} T^{48} - 909807503167401174 p^{16} T^{50} + 19305682322219015 p^{18} T^{52} - 365572082566210 p^{20} T^{54} + 6092118022573 p^{22} T^{56} - 87680564496 p^{24} T^{58} + 1061535793 p^{26} T^{60} - 452078 p^{29} T^{62} + 77399 p^{30} T^{64} - 390 p^{32} T^{66} + p^{34} T^{68} \)
29 \( ( 1 + 8 T + 280 T^{2} + 1986 T^{3} + 38495 T^{4} + 249884 T^{5} + 120677 p T^{6} + 21143382 T^{7} + 237233865 T^{8} + 1346718616 T^{9} + 12796764223 T^{10} + 68628753698 T^{11} + 572314997735 T^{12} + 2906023079268 T^{13} + 21820589877857 T^{14} + 104770719965990 T^{15} + 722244133029203 T^{16} + 3261908176831296 T^{17} + 722244133029203 p T^{18} + 104770719965990 p^{2} T^{19} + 21820589877857 p^{3} T^{20} + 2906023079268 p^{4} T^{21} + 572314997735 p^{5} T^{22} + 68628753698 p^{6} T^{23} + 12796764223 p^{7} T^{24} + 1346718616 p^{8} T^{25} + 237233865 p^{9} T^{26} + 21143382 p^{10} T^{27} + 120677 p^{12} T^{28} + 249884 p^{12} T^{29} + 38495 p^{13} T^{30} + 1986 p^{14} T^{31} + 280 p^{15} T^{32} + 8 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
31 \( ( 1 - 2 T + 274 T^{2} - 438 T^{3} + 38283 T^{4} - 49456 T^{5} + 3629113 T^{6} - 3882350 T^{7} + 261988246 T^{8} - 239600890 T^{9} + 15286592715 T^{10} - 12308004016 T^{11} + 745422068666 T^{12} - 539717985316 T^{13} + 30972070210148 T^{14} - 20475487784252 T^{15} + 35764641776066 p T^{16} - 678116696787776 T^{17} + 35764641776066 p^{2} T^{18} - 20475487784252 p^{2} T^{19} + 30972070210148 p^{3} T^{20} - 539717985316 p^{4} T^{21} + 745422068666 p^{5} T^{22} - 12308004016 p^{6} T^{23} + 15286592715 p^{7} T^{24} - 239600890 p^{8} T^{25} + 261988246 p^{9} T^{26} - 3882350 p^{10} T^{27} + 3629113 p^{11} T^{28} - 49456 p^{12} T^{29} + 38283 p^{13} T^{30} - 438 p^{14} T^{31} + 274 p^{15} T^{32} - 2 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
37 \( ( 1 - 6 T + 291 T^{2} - 1114 T^{3} + 39201 T^{4} - 94944 T^{5} + 3531997 T^{6} - 5513746 T^{7} + 247356363 T^{8} - 263578318 T^{9} + 14354144765 T^{10} - 12006343760 T^{11} + 722409688070 T^{12} - 587638275932 T^{13} + 32491496385510 T^{14} - 27826635034132 T^{15} + 1320360130234934 T^{16} - 1125433059204576 T^{17} + 1320360130234934 p T^{18} - 27826635034132 p^{2} T^{19} + 32491496385510 p^{3} T^{20} - 587638275932 p^{4} T^{21} + 722409688070 p^{5} T^{22} - 12006343760 p^{6} T^{23} + 14354144765 p^{7} T^{24} - 263578318 p^{8} T^{25} + 247356363 p^{9} T^{26} - 5513746 p^{10} T^{27} + 3531997 p^{11} T^{28} - 94944 p^{12} T^{29} + 39201 p^{13} T^{30} - 1114 p^{14} T^{31} + 291 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
41 \( 1 - 800 T^{2} + 318522 T^{4} - 84180208 T^{6} + 16613975493 T^{8} - 2611646101660 T^{10} + 340551337922989 T^{12} - 37880873391773140 T^{14} + 3668463048471344246 T^{16} - \)\(31\!\cdots\!76\)\( T^{18} + \)\(58\!\cdots\!57\)\( p T^{20} - \)\(16\!\cdots\!68\)\( T^{22} + \)\(10\!\cdots\!74\)\( T^{24} - \)\(60\!\cdots\!88\)\( T^{26} + \)\(32\!\cdots\!56\)\( T^{28} - \)\(16\!\cdots\!04\)\( T^{30} + \)\(73\!\cdots\!46\)\( T^{32} - \)\(31\!\cdots\!52\)\( T^{34} + \)\(73\!\cdots\!46\)\( p^{2} T^{36} - \)\(16\!\cdots\!04\)\( p^{4} T^{38} + \)\(32\!\cdots\!56\)\( p^{6} T^{40} - \)\(60\!\cdots\!88\)\( p^{8} T^{42} + \)\(10\!\cdots\!74\)\( p^{10} T^{44} - \)\(16\!\cdots\!68\)\( p^{12} T^{46} + \)\(58\!\cdots\!57\)\( p^{15} T^{48} - \)\(31\!\cdots\!76\)\( p^{16} T^{50} + 3668463048471344246 p^{18} T^{52} - 37880873391773140 p^{20} T^{54} + 340551337922989 p^{22} T^{56} - 2611646101660 p^{24} T^{58} + 16613975493 p^{26} T^{60} - 84180208 p^{28} T^{62} + 318522 p^{30} T^{64} - 800 p^{32} T^{66} + p^{34} T^{68} \)
43 \( ( 1 - 2 T + 416 T^{2} - 758 T^{3} + 84891 T^{4} - 141632 T^{5} + 11379069 T^{6} - 17778238 T^{7} + 1133402672 T^{8} - 1713556938 T^{9} + 89941328439 T^{10} - 135325872864 T^{11} + 5940357945610 T^{12} - 8972862446772 T^{13} + 335639372365200 T^{14} - 501267558316604 T^{15} + 16484044351581038 T^{16} - 23531265769512896 T^{17} + 16484044351581038 p T^{18} - 501267558316604 p^{2} T^{19} + 335639372365200 p^{3} T^{20} - 8972862446772 p^{4} T^{21} + 5940357945610 p^{5} T^{22} - 135325872864 p^{6} T^{23} + 89941328439 p^{7} T^{24} - 1713556938 p^{8} T^{25} + 1133402672 p^{9} T^{26} - 17778238 p^{10} T^{27} + 11379069 p^{11} T^{28} - 141632 p^{12} T^{29} + 84891 p^{13} T^{30} - 758 p^{14} T^{31} + 416 p^{15} T^{32} - 2 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
47 \( 1 - 786 T^{2} + 305642 T^{4} - 78650116 T^{6} + 15125906839 T^{8} - 2328618539216 T^{10} + 300069017373105 T^{12} - 33392461272792812 T^{14} + 3282424429226047787 T^{16} - \)\(28\!\cdots\!56\)\( T^{18} + \)\(23\!\cdots\!05\)\( T^{20} - \)\(17\!\cdots\!12\)\( T^{22} + \)\(11\!\cdots\!71\)\( T^{24} - \)\(73\!\cdots\!04\)\( T^{26} + \)\(43\!\cdots\!69\)\( T^{28} - \)\(24\!\cdots\!64\)\( T^{30} + \)\(12\!\cdots\!77\)\( T^{32} - \)\(60\!\cdots\!48\)\( T^{34} + \)\(12\!\cdots\!77\)\( p^{2} T^{36} - \)\(24\!\cdots\!64\)\( p^{4} T^{38} + \)\(43\!\cdots\!69\)\( p^{6} T^{40} - \)\(73\!\cdots\!04\)\( p^{8} T^{42} + \)\(11\!\cdots\!71\)\( p^{10} T^{44} - \)\(17\!\cdots\!12\)\( p^{12} T^{46} + \)\(23\!\cdots\!05\)\( p^{14} T^{48} - \)\(28\!\cdots\!56\)\( p^{16} T^{50} + 3282424429226047787 p^{18} T^{52} - 33392461272792812 p^{20} T^{54} + 300069017373105 p^{22} T^{56} - 2328618539216 p^{24} T^{58} + 15125906839 p^{26} T^{60} - 78650116 p^{28} T^{62} + 305642 p^{30} T^{64} - 786 p^{32} T^{66} + p^{34} T^{68} \)
53 \( 1 - 1088 T^{2} + 593138 T^{4} - 215770592 T^{6} + 58848845469 T^{8} - 12818858766636 T^{10} + 2319881866245933 T^{12} - 358273125326199780 T^{14} + 48130466985201250542 T^{16} - \)\(57\!\cdots\!24\)\( T^{18} + \)\(60\!\cdots\!65\)\( T^{20} - \)\(10\!\cdots\!84\)\( p T^{22} + \)\(49\!\cdots\!38\)\( T^{24} - \)\(38\!\cdots\!28\)\( T^{26} + \)\(27\!\cdots\!32\)\( T^{28} - \)\(18\!\cdots\!08\)\( T^{30} + \)\(10\!\cdots\!98\)\( T^{32} - \)\(60\!\cdots\!84\)\( T^{34} + \)\(10\!\cdots\!98\)\( p^{2} T^{36} - \)\(18\!\cdots\!08\)\( p^{4} T^{38} + \)\(27\!\cdots\!32\)\( p^{6} T^{40} - \)\(38\!\cdots\!28\)\( p^{8} T^{42} + \)\(49\!\cdots\!38\)\( p^{10} T^{44} - \)\(10\!\cdots\!84\)\( p^{13} T^{46} + \)\(60\!\cdots\!65\)\( p^{14} T^{48} - \)\(57\!\cdots\!24\)\( p^{16} T^{50} + 48130466985201250542 p^{18} T^{52} - 358273125326199780 p^{20} T^{54} + 2319881866245933 p^{22} T^{56} - 12818858766636 p^{24} T^{58} + 58848845469 p^{26} T^{60} - 215770592 p^{28} T^{62} + 593138 p^{30} T^{64} - 1088 p^{32} T^{66} + p^{34} T^{68} \)
59 \( 1 - 862 T^{2} + 368095 T^{4} - 103746158 T^{6} + 21709052577 T^{8} - 3601259631780 T^{10} + 494763257409217 T^{12} - 58212540248491374 T^{14} + 6035523432645758179 T^{16} - \)\(56\!\cdots\!02\)\( T^{18} + \)\(48\!\cdots\!13\)\( T^{20} - \)\(39\!\cdots\!16\)\( T^{22} + \)\(30\!\cdots\!14\)\( T^{24} - \)\(22\!\cdots\!88\)\( T^{26} + \)\(15\!\cdots\!62\)\( T^{28} - \)\(10\!\cdots\!16\)\( T^{30} + \)\(64\!\cdots\!50\)\( T^{32} - \)\(38\!\cdots\!84\)\( T^{34} + \)\(64\!\cdots\!50\)\( p^{2} T^{36} - \)\(10\!\cdots\!16\)\( p^{4} T^{38} + \)\(15\!\cdots\!62\)\( p^{6} T^{40} - \)\(22\!\cdots\!88\)\( p^{8} T^{42} + \)\(30\!\cdots\!14\)\( p^{10} T^{44} - \)\(39\!\cdots\!16\)\( p^{12} T^{46} + \)\(48\!\cdots\!13\)\( p^{14} T^{48} - \)\(56\!\cdots\!02\)\( p^{16} T^{50} + 6035523432645758179 p^{18} T^{52} - 58212540248491374 p^{20} T^{54} + 494763257409217 p^{22} T^{56} - 3601259631780 p^{24} T^{58} + 21709052577 p^{26} T^{60} - 103746158 p^{28} T^{62} + 368095 p^{30} T^{64} - 862 p^{32} T^{66} + p^{34} T^{68} \)
61 \( 1 - 1306 T^{2} + 846889 T^{4} - 363644440 T^{6} + 116320882280 T^{8} - 29561600710304 T^{10} + 6215376728479224 T^{12} - 1111435431311189480 T^{14} + \)\(17\!\cdots\!16\)\( T^{16} - \)\(23\!\cdots\!00\)\( T^{18} + \)\(28\!\cdots\!40\)\( T^{20} - \)\(31\!\cdots\!68\)\( T^{22} + \)\(31\!\cdots\!12\)\( T^{24} - \)\(27\!\cdots\!16\)\( T^{26} + \)\(22\!\cdots\!12\)\( T^{28} - \)\(17\!\cdots\!92\)\( T^{30} + \)\(11\!\cdots\!62\)\( T^{32} - \)\(76\!\cdots\!20\)\( T^{34} + \)\(11\!\cdots\!62\)\( p^{2} T^{36} - \)\(17\!\cdots\!92\)\( p^{4} T^{38} + \)\(22\!\cdots\!12\)\( p^{6} T^{40} - \)\(27\!\cdots\!16\)\( p^{8} T^{42} + \)\(31\!\cdots\!12\)\( p^{10} T^{44} - \)\(31\!\cdots\!68\)\( p^{12} T^{46} + \)\(28\!\cdots\!40\)\( p^{14} T^{48} - \)\(23\!\cdots\!00\)\( p^{16} T^{50} + \)\(17\!\cdots\!16\)\( p^{18} T^{52} - 1111435431311189480 p^{20} T^{54} + 6215376728479224 p^{22} T^{56} - 29561600710304 p^{24} T^{58} + 116320882280 p^{26} T^{60} - 363644440 p^{28} T^{62} + 846889 p^{30} T^{64} - 1306 p^{32} T^{66} + p^{34} T^{68} \)
71 \( 1 - 1056 T^{2} + 549229 T^{4} - 188159736 T^{6} + 47963906492 T^{8} - 9754150784304 T^{10} + 1658359742442852 T^{12} - 243960464630288360 T^{14} + 31880216274224415792 T^{16} - \)\(37\!\cdots\!40\)\( T^{18} + \)\(41\!\cdots\!80\)\( T^{20} - \)\(41\!\cdots\!72\)\( T^{22} + \)\(39\!\cdots\!04\)\( T^{24} - \)\(35\!\cdots\!08\)\( T^{26} + \)\(30\!\cdots\!48\)\( T^{28} - \)\(24\!\cdots\!88\)\( T^{30} + \)\(18\!\cdots\!70\)\( T^{32} - \)\(13\!\cdots\!96\)\( T^{34} + \)\(18\!\cdots\!70\)\( p^{2} T^{36} - \)\(24\!\cdots\!88\)\( p^{4} T^{38} + \)\(30\!\cdots\!48\)\( p^{6} T^{40} - \)\(35\!\cdots\!08\)\( p^{8} T^{42} + \)\(39\!\cdots\!04\)\( p^{10} T^{44} - \)\(41\!\cdots\!72\)\( p^{12} T^{46} + \)\(41\!\cdots\!80\)\( p^{14} T^{48} - \)\(37\!\cdots\!40\)\( p^{16} T^{50} + 31880216274224415792 p^{18} T^{52} - 243960464630288360 p^{20} T^{54} + 1658359742442852 p^{22} T^{56} - 9754150784304 p^{24} T^{58} + 47963906492 p^{26} T^{60} - 188159736 p^{28} T^{62} + 549229 p^{30} T^{64} - 1056 p^{32} T^{66} + p^{34} T^{68} \)
73 \( ( 1 - 6 T + 515 T^{2} - 3328 T^{3} + 144541 T^{4} - 940110 T^{5} + 28581217 T^{6} - 180185960 T^{7} + 4391332191 T^{8} - 26455644678 T^{9} + 553385774101 T^{10} - 3166334384520 T^{11} + 59248572421426 T^{12} - 321034920001700 T^{13} + 5516399267831014 T^{14} - 28277250519852264 T^{15} + 453126281475821258 T^{16} - 2194400317569554068 T^{17} + 453126281475821258 p T^{18} - 28277250519852264 p^{2} T^{19} + 5516399267831014 p^{3} T^{20} - 321034920001700 p^{4} T^{21} + 59248572421426 p^{5} T^{22} - 3166334384520 p^{6} T^{23} + 553385774101 p^{7} T^{24} - 26455644678 p^{8} T^{25} + 4391332191 p^{9} T^{26} - 180185960 p^{10} T^{27} + 28581217 p^{11} T^{28} - 940110 p^{12} T^{29} + 144541 p^{13} T^{30} - 3328 p^{14} T^{31} + 515 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
79 \( ( 1 - 6 T + 865 T^{2} - 4056 T^{3} + 364802 T^{4} - 1343440 T^{5} + 100925602 T^{6} - 294801704 T^{7} + 20670590462 T^{8} - 48642581736 T^{9} + 3336713662750 T^{10} - 6461771515768 T^{11} + 440013510687962 T^{12} - 720657077267376 T^{13} + 48426256361249146 T^{14} - 69315757086640520 T^{15} + 4502866981735763504 T^{16} - 5837866981735029028 T^{17} + 4502866981735763504 p T^{18} - 69315757086640520 p^{2} T^{19} + 48426256361249146 p^{3} T^{20} - 720657077267376 p^{4} T^{21} + 440013510687962 p^{5} T^{22} - 6461771515768 p^{6} T^{23} + 3336713662750 p^{7} T^{24} - 48642581736 p^{8} T^{25} + 20670590462 p^{9} T^{26} - 294801704 p^{10} T^{27} + 100925602 p^{11} T^{28} - 1343440 p^{12} T^{29} + 364802 p^{13} T^{30} - 4056 p^{14} T^{31} + 865 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
83 \( 1 - 1292 T^{2} + 848290 T^{4} - 376704388 T^{6} + 127097108205 T^{8} - 34708595206900 T^{10} + 7983013214105213 T^{12} - 1589072191555694912 T^{14} + \)\(27\!\cdots\!38\)\( T^{16} - \)\(43\!\cdots\!24\)\( T^{18} + \)\(62\!\cdots\!01\)\( T^{20} - \)\(81\!\cdots\!72\)\( T^{22} + \)\(98\!\cdots\!82\)\( T^{24} - \)\(11\!\cdots\!68\)\( T^{26} + \)\(11\!\cdots\!84\)\( T^{28} - \)\(11\!\cdots\!36\)\( T^{30} + \)\(10\!\cdots\!30\)\( T^{32} - \)\(86\!\cdots\!56\)\( T^{34} + \)\(10\!\cdots\!30\)\( p^{2} T^{36} - \)\(11\!\cdots\!36\)\( p^{4} T^{38} + \)\(11\!\cdots\!84\)\( p^{6} T^{40} - \)\(11\!\cdots\!68\)\( p^{8} T^{42} + \)\(98\!\cdots\!82\)\( p^{10} T^{44} - \)\(81\!\cdots\!72\)\( p^{12} T^{46} + \)\(62\!\cdots\!01\)\( p^{14} T^{48} - \)\(43\!\cdots\!24\)\( p^{16} T^{50} + \)\(27\!\cdots\!38\)\( p^{18} T^{52} - 1589072191555694912 p^{20} T^{54} + 7983013214105213 p^{22} T^{56} - 34708595206900 p^{24} T^{58} + 127097108205 p^{26} T^{60} - 376704388 p^{28} T^{62} + 848290 p^{30} T^{64} - 1292 p^{32} T^{66} + p^{34} T^{68} \)
89 \( ( 1 + 638 T^{2} - 1230 T^{3} + 210827 T^{4} - 839464 T^{5} + 48519761 T^{6} - 283155618 T^{7} + 8787644647 T^{8} - 64303304896 T^{9} + 1328098248333 T^{10} - 11104840293638 T^{11} + 172455439401115 T^{12} - 1544526529348152 T^{13} + 19580679895915225 T^{14} - 178000703040889418 T^{15} + 1964090161809709429 T^{16} - 17240520658784583360 T^{17} + 1964090161809709429 p T^{18} - 178000703040889418 p^{2} T^{19} + 19580679895915225 p^{3} T^{20} - 1544526529348152 p^{4} T^{21} + 172455439401115 p^{5} T^{22} - 11104840293638 p^{6} T^{23} + 1328098248333 p^{7} T^{24} - 64303304896 p^{8} T^{25} + 8787644647 p^{9} T^{26} - 283155618 p^{10} T^{27} + 48519761 p^{11} T^{28} - 839464 p^{12} T^{29} + 210827 p^{13} T^{30} - 1230 p^{14} T^{31} + 638 p^{15} T^{32} + p^{17} T^{34} )^{2} \)
97 \( 1 - 1830 T^{2} + 1694021 T^{4} - 1054854944 T^{6} + 495970293140 T^{8} - 187442459787088 T^{10} + 59203185437163852 T^{12} - 16044756844465501184 T^{14} + \)\(38\!\cdots\!88\)\( T^{16} - \)\(82\!\cdots\!48\)\( p T^{18} + \)\(15\!\cdots\!44\)\( T^{20} - \)\(25\!\cdots\!48\)\( T^{22} + \)\(39\!\cdots\!80\)\( T^{24} - \)\(55\!\cdots\!76\)\( T^{26} + \)\(72\!\cdots\!76\)\( T^{28} - \)\(85\!\cdots\!52\)\( T^{30} + \)\(94\!\cdots\!86\)\( T^{32} - \)\(95\!\cdots\!44\)\( T^{34} + \)\(94\!\cdots\!86\)\( p^{2} T^{36} - \)\(85\!\cdots\!52\)\( p^{4} T^{38} + \)\(72\!\cdots\!76\)\( p^{6} T^{40} - \)\(55\!\cdots\!76\)\( p^{8} T^{42} + \)\(39\!\cdots\!80\)\( p^{10} T^{44} - \)\(25\!\cdots\!48\)\( p^{12} T^{46} + \)\(15\!\cdots\!44\)\( p^{14} T^{48} - \)\(82\!\cdots\!48\)\( p^{17} T^{50} + \)\(38\!\cdots\!88\)\( p^{18} T^{52} - 16044756844465501184 p^{20} T^{54} + 59203185437163852 p^{22} T^{56} - 187442459787088 p^{24} T^{58} + 495970293140 p^{26} T^{60} - 1054854944 p^{28} T^{62} + 1694021 p^{30} T^{64} - 1830 p^{32} T^{66} + p^{34} T^{68} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{68} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.41209145249051837638126203132, −1.31411386668856942105846531027, −1.28721531746563933478051822696, −1.23687518462739566124161942694, −1.13460047968296666964669883669, −1.12925062479673683430958988037, −1.09902493018312668013453133400, −1.08917398874196517166722722911, −1.07268102807469424171602819577, −1.05778993123057833193106108980, −1.03113108402797166826363700924, −0.942458577335231169673621104163, −0.852951875550104829337650171678, −0.814099668160534009859036020399, −0.75672055331039944556574291531, −0.75408836983968483008820270384, −0.69019199637017262044515562250, −0.41396006874749833423698679929, −0.39159835404168123447899487524, −0.31042252254433395966927619500, −0.29612455011257875637117248644, −0.28036372853586026110305115775, −0.26991393292343422336989653963, −0.25679119546566664696349043941, −0.14761263995010301939642637944, 0.14761263995010301939642637944, 0.25679119546566664696349043941, 0.26991393292343422336989653963, 0.28036372853586026110305115775, 0.29612455011257875637117248644, 0.31042252254433395966927619500, 0.39159835404168123447899487524, 0.41396006874749833423698679929, 0.69019199637017262044515562250, 0.75408836983968483008820270384, 0.75672055331039944556574291531, 0.814099668160534009859036020399, 0.852951875550104829337650171678, 0.942458577335231169673621104163, 1.03113108402797166826363700924, 1.05778993123057833193106108980, 1.07268102807469424171602819577, 1.08917398874196517166722722911, 1.09902493018312668013453133400, 1.12925062479673683430958988037, 1.13460047968296666964669883669, 1.23687518462739566124161942694, 1.28721531746563933478051822696, 1.31411386668856942105846531027, 1.41209145249051837638126203132

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.