Properties

Degree 28
Conductor $ 2^{28} \cdot 3^{14} \cdot 5^{14} \cdot 67^{14} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·3-s + 14·5-s + 3·7-s + 105·9-s + 6·11-s + 9·13-s − 196·15-s − 12·17-s + 14·19-s − 42·21-s + 6·23-s + 105·25-s − 560·27-s − 29-s + 7·31-s − 84·33-s + 42·35-s − 2·37-s − 126·39-s − 18·41-s − 6·43-s + 1.47e3·45-s + 7·47-s + 29·49-s + 168·51-s − 12·53-s + 84·55-s + ⋯
L(s)  = 1  − 8.08·3-s + 6.26·5-s + 1.13·7-s + 35·9-s + 1.80·11-s + 2.49·13-s − 50.6·15-s − 2.91·17-s + 3.21·19-s − 9.16·21-s + 1.25·23-s + 21·25-s − 107.·27-s − 0.185·29-s + 1.25·31-s − 14.6·33-s + 7.09·35-s − 0.328·37-s − 20.1·39-s − 2.81·41-s − 0.914·43-s + 219.·45-s + 1.02·47-s + 29/7·49-s + 23.5·51-s − 1.64·53-s + 11.3·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14} \cdot 5^{14} \cdot 67^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14} \cdot 5^{14} \cdot 67^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(28\)
\( N \)  =  \(2^{28} \cdot 3^{14} \cdot 5^{14} \cdot 67^{14}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4020} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(28,\ 2^{28} \cdot 3^{14} \cdot 5^{14} \cdot 67^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )$
$L(1)$  $\approx$  $61.64584349$
$L(\frac12)$  $\approx$  $61.64584349$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \(F_p\) is a polynomial of degree 28. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 27.
$p$$F_p$
bad2 \( 1 \)
3 \( ( 1 + T )^{14} \)
5 \( ( 1 - T )^{14} \)
67 \( 1 - 25 T + 298 T^{2} - 2382 T^{3} + 12789 T^{4} - 5772 T^{5} - 884312 T^{6} + 10995854 T^{7} - 884312 p T^{8} - 5772 p^{2} T^{9} + 12789 p^{3} T^{10} - 2382 p^{4} T^{11} + 298 p^{5} T^{12} - 25 p^{6} T^{13} + p^{7} T^{14} \)
good7 \( 1 - 3 T - 20 T^{2} + 57 T^{3} + 177 T^{4} - 389 T^{5} - 1803 T^{6} + 3303 T^{7} + 18070 T^{8} - 36377 T^{9} - 110178 T^{10} + 123265 T^{11} + 781213 T^{12} + 114570 T^{13} - 6762664 T^{14} + 114570 p T^{15} + 781213 p^{2} T^{16} + 123265 p^{3} T^{17} - 110178 p^{4} T^{18} - 36377 p^{5} T^{19} + 18070 p^{6} T^{20} + 3303 p^{7} T^{21} - 1803 p^{8} T^{22} - 389 p^{9} T^{23} + 177 p^{10} T^{24} + 57 p^{11} T^{25} - 20 p^{12} T^{26} - 3 p^{13} T^{27} + p^{14} T^{28} \)
11 \( 1 - 6 T - p T^{2} + 16 T^{3} + 50 p T^{4} + 41 p T^{5} - 6158 T^{6} - 23818 T^{7} + 26095 T^{8} + 323347 T^{9} + 407400 T^{10} - 2104801 T^{11} - 9984167 T^{12} + 1347259 p T^{13} + 70409980 T^{14} + 1347259 p^{2} T^{15} - 9984167 p^{2} T^{16} - 2104801 p^{3} T^{17} + 407400 p^{4} T^{18} + 323347 p^{5} T^{19} + 26095 p^{6} T^{20} - 23818 p^{7} T^{21} - 6158 p^{8} T^{22} + 41 p^{10} T^{23} + 50 p^{11} T^{24} + 16 p^{11} T^{25} - p^{13} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \)
13 \( 1 - 9 T - 3 T^{2} + 18 p T^{3} - 216 T^{4} - 3440 T^{5} + 5289 T^{6} + 22909 T^{7} - 40116 T^{8} + 156307 T^{9} - 627413 T^{10} - 5854224 T^{11} + 29555027 T^{12} + 38130675 T^{13} - 498920098 T^{14} + 38130675 p T^{15} + 29555027 p^{2} T^{16} - 5854224 p^{3} T^{17} - 627413 p^{4} T^{18} + 156307 p^{5} T^{19} - 40116 p^{6} T^{20} + 22909 p^{7} T^{21} + 5289 p^{8} T^{22} - 3440 p^{9} T^{23} - 216 p^{10} T^{24} + 18 p^{12} T^{25} - 3 p^{12} T^{26} - 9 p^{13} T^{27} + p^{14} T^{28} \)
17 \( 1 + 12 T + 13 T^{2} - 120 T^{3} + 1763 T^{4} + 8439 T^{5} - 29854 T^{6} + 48731 T^{7} + 1353310 T^{8} - 587405 T^{9} - 5786839 T^{10} + 84344359 T^{11} + 106817487 T^{12} - 21465324 T^{13} + 3964551862 T^{14} - 21465324 p T^{15} + 106817487 p^{2} T^{16} + 84344359 p^{3} T^{17} - 5786839 p^{4} T^{18} - 587405 p^{5} T^{19} + 1353310 p^{6} T^{20} + 48731 p^{7} T^{21} - 29854 p^{8} T^{22} + 8439 p^{9} T^{23} + 1763 p^{10} T^{24} - 120 p^{11} T^{25} + 13 p^{12} T^{26} + 12 p^{13} T^{27} + p^{14} T^{28} \)
19 \( 1 - 14 T + 8 T^{2} + 582 T^{3} + 222 T^{4} - 25358 T^{5} - 8233 T^{6} + 614419 T^{7} + 102013 p T^{8} - 16399564 T^{9} - 65078281 T^{10} + 228533498 T^{11} + 2092709725 T^{12} - 2476014457 T^{13} - 40027234346 T^{14} - 2476014457 p T^{15} + 2092709725 p^{2} T^{16} + 228533498 p^{3} T^{17} - 65078281 p^{4} T^{18} - 16399564 p^{5} T^{19} + 102013 p^{7} T^{20} + 614419 p^{7} T^{21} - 8233 p^{8} T^{22} - 25358 p^{9} T^{23} + 222 p^{10} T^{24} + 582 p^{11} T^{25} + 8 p^{12} T^{26} - 14 p^{13} T^{27} + p^{14} T^{28} \)
23 \( 1 - 6 T - 55 T^{2} + 462 T^{3} + 753 T^{4} - 13307 T^{5} + 10046 T^{6} + 152185 T^{7} - 95636 T^{8} - 824917 T^{9} - 12768145 T^{10} + 2239877 p T^{11} + 264832895 T^{12} - 956116592 T^{13} - 2075704102 T^{14} - 956116592 p T^{15} + 264832895 p^{2} T^{16} + 2239877 p^{4} T^{17} - 12768145 p^{4} T^{18} - 824917 p^{5} T^{19} - 95636 p^{6} T^{20} + 152185 p^{7} T^{21} + 10046 p^{8} T^{22} - 13307 p^{9} T^{23} + 753 p^{10} T^{24} + 462 p^{11} T^{25} - 55 p^{12} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \)
29 \( 1 + T - 94 T^{2} + 111 T^{3} + 4141 T^{4} - 11695 T^{5} - 81857 T^{6} + 347769 T^{7} + 213802 T^{8} + 2744015 T^{9} + 736264 T^{10} - 378858495 T^{11} + 1902063729 T^{12} + 6351771646 T^{13} - 90593298388 T^{14} + 6351771646 p T^{15} + 1902063729 p^{2} T^{16} - 378858495 p^{3} T^{17} + 736264 p^{4} T^{18} + 2744015 p^{5} T^{19} + 213802 p^{6} T^{20} + 347769 p^{7} T^{21} - 81857 p^{8} T^{22} - 11695 p^{9} T^{23} + 4141 p^{10} T^{24} + 111 p^{11} T^{25} - 94 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \)
31 \( 1 - 7 T - 39 T^{2} + 660 T^{3} - 2351 T^{4} - 13913 T^{5} + 146821 T^{6} - 144326 T^{7} - 3076282 T^{8} + 12618238 T^{9} + 70496210 T^{10} - 862571403 T^{11} + 1968366689 T^{12} + 19721322013 T^{13} - 189458995778 T^{14} + 19721322013 p T^{15} + 1968366689 p^{2} T^{16} - 862571403 p^{3} T^{17} + 70496210 p^{4} T^{18} + 12618238 p^{5} T^{19} - 3076282 p^{6} T^{20} - 144326 p^{7} T^{21} + 146821 p^{8} T^{22} - 13913 p^{9} T^{23} - 2351 p^{10} T^{24} + 660 p^{11} T^{25} - 39 p^{12} T^{26} - 7 p^{13} T^{27} + p^{14} T^{28} \)
37 \( 1 + 2 T - 144 T^{2} - 58 T^{3} + 11562 T^{4} - 8564 T^{5} - 567933 T^{6} + 1219349 T^{7} + 16972491 T^{8} - 72220508 T^{9} - 82031967 T^{10} + 2604100054 T^{11} - 19609237189 T^{12} - 38834898747 T^{13} + 1110997548598 T^{14} - 38834898747 p T^{15} - 19609237189 p^{2} T^{16} + 2604100054 p^{3} T^{17} - 82031967 p^{4} T^{18} - 72220508 p^{5} T^{19} + 16972491 p^{6} T^{20} + 1219349 p^{7} T^{21} - 567933 p^{8} T^{22} - 8564 p^{9} T^{23} + 11562 p^{10} T^{24} - 58 p^{11} T^{25} - 144 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \)
41 \( 1 + 18 T - 29 T^{2} - 2200 T^{3} - 1130 T^{4} + 183989 T^{5} + 379300 T^{6} - 8058104 T^{7} - 4062155 T^{8} + 263971859 T^{9} - 1279880946 T^{10} - 2948011895 T^{11} + 160106068045 T^{12} + 57760123657 T^{13} - 7992127618244 T^{14} + 57760123657 p T^{15} + 160106068045 p^{2} T^{16} - 2948011895 p^{3} T^{17} - 1279880946 p^{4} T^{18} + 263971859 p^{5} T^{19} - 4062155 p^{6} T^{20} - 8058104 p^{7} T^{21} + 379300 p^{8} T^{22} + 183989 p^{9} T^{23} - 1130 p^{10} T^{24} - 2200 p^{11} T^{25} - 29 p^{12} T^{26} + 18 p^{13} T^{27} + p^{14} T^{28} \)
43 \( ( 1 + 3 T + 165 T^{2} + 625 T^{3} + 14336 T^{4} + 1330 p T^{5} + 833372 T^{6} + 3148724 T^{7} + 833372 p T^{8} + 1330 p^{3} T^{9} + 14336 p^{3} T^{10} + 625 p^{4} T^{11} + 165 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
47 \( 1 - 7 T - 153 T^{2} + 1362 T^{3} + 9935 T^{4} - 114473 T^{5} - 354685 T^{6} + 5522720 T^{7} + 11215060 T^{8} - 156793834 T^{9} - 900317666 T^{10} + 2422525493 T^{11} + 78249322737 T^{12} - 362708337 p T^{13} - 95369553782 p T^{14} - 362708337 p^{2} T^{15} + 78249322737 p^{2} T^{16} + 2422525493 p^{3} T^{17} - 900317666 p^{4} T^{18} - 156793834 p^{5} T^{19} + 11215060 p^{6} T^{20} + 5522720 p^{7} T^{21} - 354685 p^{8} T^{22} - 114473 p^{9} T^{23} + 9935 p^{10} T^{24} + 1362 p^{11} T^{25} - 153 p^{12} T^{26} - 7 p^{13} T^{27} + p^{14} T^{28} \)
53 \( ( 1 + 6 T + 247 T^{2} + 1145 T^{3} + 30041 T^{4} + 116883 T^{5} + 2344387 T^{6} + 7652092 T^{7} + 2344387 p T^{8} + 116883 p^{2} T^{9} + 30041 p^{3} T^{10} + 1145 p^{4} T^{11} + 247 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
59 \( ( 1 + T + 289 T^{2} + 403 T^{3} + 39692 T^{4} + 1076 p T^{5} + 57860 p T^{6} + 5088680 T^{7} + 57860 p^{2} T^{8} + 1076 p^{3} T^{9} + 39692 p^{3} T^{10} + 403 p^{4} T^{11} + 289 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} )^{2} \)
61 \( 1 - 250 T^{2} - 1082 T^{3} + 35548 T^{4} + 260866 T^{5} - 2717433 T^{6} - 36918565 T^{7} + 78765775 T^{8} + 3261082584 T^{9} + 10157211285 T^{10} - 189570224500 T^{11} - 1612043826243 T^{12} + 4675152702057 T^{13} + 127013861698650 T^{14} + 4675152702057 p T^{15} - 1612043826243 p^{2} T^{16} - 189570224500 p^{3} T^{17} + 10157211285 p^{4} T^{18} + 3261082584 p^{5} T^{19} + 78765775 p^{6} T^{20} - 36918565 p^{7} T^{21} - 2717433 p^{8} T^{22} + 260866 p^{9} T^{23} + 35548 p^{10} T^{24} - 1082 p^{11} T^{25} - 250 p^{12} T^{26} + p^{14} T^{28} \)
71 \( 1 - 22 T - 17 T^{2} + 820 T^{3} + 49824 T^{4} - 364769 T^{5} - 2625066 T^{6} - 31439174 T^{7} + 604960277 T^{8} + 1291494445 T^{9} + 13047003346 T^{10} - 519002271697 T^{11} - 392108245783 T^{12} - 2884405046697 T^{13} + 359176316454548 T^{14} - 2884405046697 p T^{15} - 392108245783 p^{2} T^{16} - 519002271697 p^{3} T^{17} + 13047003346 p^{4} T^{18} + 1291494445 p^{5} T^{19} + 604960277 p^{6} T^{20} - 31439174 p^{7} T^{21} - 2625066 p^{8} T^{22} - 364769 p^{9} T^{23} + 49824 p^{10} T^{24} + 820 p^{11} T^{25} - 17 p^{12} T^{26} - 22 p^{13} T^{27} + p^{14} T^{28} \)
73 \( 1 + 15 T - 305 T^{2} - 3966 T^{3} + 75355 T^{4} + 692695 T^{5} - 13270379 T^{6} - 81309244 T^{7} + 1857676842 T^{8} + 7051146754 T^{9} - 209843950998 T^{10} - 423003183473 T^{11} + 19720732477631 T^{12} + 12001986433031 T^{13} - 1563096084841678 T^{14} + 12001986433031 p T^{15} + 19720732477631 p^{2} T^{16} - 423003183473 p^{3} T^{17} - 209843950998 p^{4} T^{18} + 7051146754 p^{5} T^{19} + 1857676842 p^{6} T^{20} - 81309244 p^{7} T^{21} - 13270379 p^{8} T^{22} + 692695 p^{9} T^{23} + 75355 p^{10} T^{24} - 3966 p^{11} T^{25} - 305 p^{12} T^{26} + 15 p^{13} T^{27} + p^{14} T^{28} \)
79 \( 1 - 9 T - 290 T^{2} + 971 T^{3} + 58957 T^{4} + 39935 T^{5} - 7304652 T^{6} - 31245424 T^{7} + 627140605 T^{8} + 4416045423 T^{9} - 33081571720 T^{10} - 353555917085 T^{11} + 956394409296 T^{12} + 11107915331417 T^{13} - 6705152277818 T^{14} + 11107915331417 p T^{15} + 956394409296 p^{2} T^{16} - 353555917085 p^{3} T^{17} - 33081571720 p^{4} T^{18} + 4416045423 p^{5} T^{19} + 627140605 p^{6} T^{20} - 31245424 p^{7} T^{21} - 7304652 p^{8} T^{22} + 39935 p^{9} T^{23} + 58957 p^{10} T^{24} + 971 p^{11} T^{25} - 290 p^{12} T^{26} - 9 p^{13} T^{27} + p^{14} T^{28} \)
83 \( 1 + T - 299 T^{2} + 1040 T^{3} + 46121 T^{4} - 338171 T^{5} - 3970157 T^{6} + 54632160 T^{7} + 144189246 T^{8} - 5524282328 T^{9} + 13760244348 T^{10} + 359721042959 T^{11} - 3019786479455 T^{12} - 11121758694339 T^{13} + 306888352958254 T^{14} - 11121758694339 p T^{15} - 3019786479455 p^{2} T^{16} + 359721042959 p^{3} T^{17} + 13760244348 p^{4} T^{18} - 5524282328 p^{5} T^{19} + 144189246 p^{6} T^{20} + 54632160 p^{7} T^{21} - 3970157 p^{8} T^{22} - 338171 p^{9} T^{23} + 46121 p^{10} T^{24} + 1040 p^{11} T^{25} - 299 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \)
89 \( ( 1 + 6 T + 509 T^{2} + 2393 T^{3} + 116955 T^{4} + 439711 T^{5} + 16028883 T^{6} + 48830780 T^{7} + 16028883 p T^{8} + 439711 p^{2} T^{9} + 116955 p^{3} T^{10} + 2393 p^{4} T^{11} + 509 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
97 \( 1 - 16 T - 305 T^{2} + 4468 T^{3} + 66659 T^{4} - 627781 T^{5} - 13361122 T^{6} + 73927187 T^{7} + 2109835734 T^{8} - 7502842347 T^{9} - 269438721123 T^{10} + 493586727995 T^{11} + 30832175504259 T^{12} - 14324354895126 T^{13} - 3192872878082918 T^{14} - 14324354895126 p T^{15} + 30832175504259 p^{2} T^{16} + 493586727995 p^{3} T^{17} - 269438721123 p^{4} T^{18} - 7502842347 p^{5} T^{19} + 2109835734 p^{6} T^{20} + 73927187 p^{7} T^{21} - 13361122 p^{8} T^{22} - 627781 p^{9} T^{23} + 66659 p^{10} T^{24} + 4468 p^{11} T^{25} - 305 p^{12} T^{26} - 16 p^{13} T^{27} + p^{14} T^{28} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.03226374180278064634066167084, −2.03074699490986513008911496682, −2.00901533297073767895410798555, −1.93817508858286494727572929380, −1.77888440389977831971128789591, −1.73594354325011122156613726747, −1.69416362419000046442642041230, −1.59656304348831158282694346180, −1.57213592032214889268937155692, −1.56155096537628912344190344109, −1.37271409756193811370178245979, −1.31071427810578543167039692682, −1.28196914077881624124264366203, −1.26604483214766713595862910451, −1.22319116109044082862061118669, −0.869432823657595475603136525370, −0.862726364644656957896429755955, −0.798963982856257673985654999283, −0.77069607701553889850816002746, −0.70308317341682448871715088178, −0.68410962984173724828145689778, −0.58635853988780008527197485354, −0.46098479019010255791456453202, −0.34316976677185336018798266741, −0.20383383770134357620869023909, 0.20383383770134357620869023909, 0.34316976677185336018798266741, 0.46098479019010255791456453202, 0.58635853988780008527197485354, 0.68410962984173724828145689778, 0.70308317341682448871715088178, 0.77069607701553889850816002746, 0.798963982856257673985654999283, 0.862726364644656957896429755955, 0.869432823657595475603136525370, 1.22319116109044082862061118669, 1.26604483214766713595862910451, 1.28196914077881624124264366203, 1.31071427810578543167039692682, 1.37271409756193811370178245979, 1.56155096537628912344190344109, 1.57213592032214889268937155692, 1.59656304348831158282694346180, 1.69416362419000046442642041230, 1.73594354325011122156613726747, 1.77888440389977831971128789591, 1.93817508858286494727572929380, 2.00901533297073767895410798555, 2.03074699490986513008911496682, 2.03226374180278064634066167084

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

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