Properties

Degree 48
Conductor $ 3^{24} \cdot 5^{24} \cdot 7^{24} $
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  − 4·3-s + 8·9-s − 8·13-s + 4·16-s − 8·25-s − 16·27-s − 16·37-s + 32·39-s − 40·43-s − 16·48-s + 32·61-s + 24·67-s + 32·73-s + 32·75-s + 45·81-s + 24·97-s − 88·103-s + 64·111-s − 64·117-s + 140·121-s + 127-s + 160·129-s + 131-s + 137-s + 139-s + 32·144-s + 149-s + ⋯
L(s)  = 1  − 2.30·3-s + 8/3·9-s − 2.21·13-s + 16-s − 8/5·25-s − 3.07·27-s − 2.63·37-s + 5.12·39-s − 6.09·43-s − 2.30·48-s + 4.09·61-s + 2.93·67-s + 3.74·73-s + 3.69·75-s + 5·81-s + 2.43·97-s − 8.67·103-s + 6.07·111-s − 5.91·117-s + 12.7·121-s + 0.0887·127-s + 14.0·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8/3·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(48\)
\( N \)  =  \(3^{24} \cdot 5^{24} \cdot 7^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(48,\ 3^{24} \cdot 5^{24} \cdot 7^{24} ,\ ( \ : [1/2]^{24} ),\ 1 )$
$L(1)$  $\approx$  $0.156845$
$L(\frac12)$  $\approx$  $0.156845$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 48. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 47.
$p$$F_p(T)$
bad3 \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 19 T^{4} - 32 T^{5} - 152 T^{6} - 308 T^{7} - 557 T^{8} - 520 T^{9} + 848 T^{10} + 3376 T^{11} + 6274 T^{12} + 3376 p T^{13} + 848 p^{2} T^{14} - 520 p^{3} T^{15} - 557 p^{4} T^{16} - 308 p^{5} T^{17} - 152 p^{6} T^{18} - 32 p^{7} T^{19} + 19 p^{8} T^{20} + 16 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
5 \( 1 + 8 T^{2} + 62 T^{4} + 392 T^{6} + 2383 T^{8} + 12848 T^{10} + 61124 T^{12} + 12848 p^{2} T^{14} + 2383 p^{4} T^{16} + 392 p^{6} T^{18} + 62 p^{8} T^{20} + 8 p^{10} T^{22} + p^{12} T^{24} \)
7 \( ( 1 + T^{4} )^{6} \)
good2 \( 1 - p^{2} T^{4} + 3 p T^{8} - 3 p^{2} T^{12} + 49 T^{16} - 121 p^{3} T^{20} + 579 p^{4} T^{24} - 121 p^{7} T^{28} + 49 p^{8} T^{32} - 3 p^{14} T^{36} + 3 p^{17} T^{40} - p^{22} T^{44} + p^{24} T^{48} \)
11 \( ( 1 - 70 T^{2} + 2559 T^{4} - 63150 T^{6} + 1171747 T^{8} - 17282492 T^{10} + 208979274 T^{12} - 17282492 p^{2} T^{14} + 1171747 p^{4} T^{16} - 63150 p^{6} T^{18} + 2559 p^{8} T^{20} - 70 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
13 \( ( 1 + 4 T + 8 T^{2} - 72 T^{3} - 389 T^{4} + 264 T^{5} + 40 p^{2} T^{6} + 29300 T^{7} - 2109 T^{8} - 315144 T^{9} - 688816 T^{10} + 2261952 T^{11} + 20440658 T^{12} + 2261952 p T^{13} - 688816 p^{2} T^{14} - 315144 p^{3} T^{15} - 2109 p^{4} T^{16} + 29300 p^{5} T^{17} + 40 p^{8} T^{18} + 264 p^{7} T^{19} - 389 p^{8} T^{20} - 72 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
17 \( 1 + 742 T^{4} + 108863 T^{8} - 22892282 T^{12} + 2052508571 T^{16} + 4431139184980 T^{20} + 1421469189205194 T^{24} + 4431139184980 p^{4} T^{28} + 2052508571 p^{8} T^{32} - 22892282 p^{12} T^{36} + 108863 p^{16} T^{40} + 742 p^{20} T^{44} + p^{24} T^{48} \)
19 \( ( 1 - 96 T^{2} + 5062 T^{4} - 188896 T^{6} + 5494655 T^{8} - 131835584 T^{10} + 2699491796 T^{12} - 131835584 p^{2} T^{14} + 5494655 p^{4} T^{16} - 188896 p^{6} T^{18} + 5062 p^{8} T^{20} - 96 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
23 \( 1 - 1364 T^{4} + 1100226 T^{8} - 29719836 p T^{12} + 378616721071 T^{16} - 173788773189544 T^{20} + 79328990254613916 T^{24} - 173788773189544 p^{4} T^{28} + 378616721071 p^{8} T^{32} - 29719836 p^{13} T^{36} + 1100226 p^{16} T^{40} - 1364 p^{20} T^{44} + p^{24} T^{48} \)
29 \( ( 1 + 134 T^{2} + 8831 T^{4} + 395910 T^{6} + 15260203 T^{8} + 553509028 T^{10} + 17643760490 T^{12} + 553509028 p^{2} T^{14} + 15260203 p^{4} T^{16} + 395910 p^{6} T^{18} + 8831 p^{8} T^{20} + 134 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
31 \( ( 1 + 108 T^{2} - 144 T^{3} + 5563 T^{4} - 12656 T^{5} + 195704 T^{6} - 12656 p T^{7} + 5563 p^{2} T^{8} - 144 p^{3} T^{9} + 108 p^{4} T^{10} + p^{6} T^{12} )^{4} \)
37 \( ( 1 + 8 T + 32 T^{2} + 440 T^{3} + 4622 T^{4} + 12056 T^{5} + 45344 T^{6} + 589416 T^{7} - 1734129 T^{8} - 37494928 T^{9} - 122030656 T^{10} - 1417591280 T^{11} - 15998825564 T^{12} - 1417591280 p T^{13} - 122030656 p^{2} T^{14} - 37494928 p^{3} T^{15} - 1734129 p^{4} T^{16} + 589416 p^{5} T^{17} + 45344 p^{6} T^{18} + 12056 p^{7} T^{19} + 4622 p^{8} T^{20} + 440 p^{9} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
41 \( ( 1 - 304 T^{2} + 41662 T^{4} - 3421264 T^{6} + 192103647 T^{8} - 8381663616 T^{10} + 338524480004 T^{12} - 8381663616 p^{2} T^{14} + 192103647 p^{4} T^{16} - 3421264 p^{6} T^{18} + 41662 p^{8} T^{20} - 304 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
43 \( ( 1 + 20 T + 200 T^{2} + 44 p T^{3} + 22578 T^{4} + 225884 T^{5} + 1791912 T^{6} + 14753484 T^{7} + 128883791 T^{8} + 968381160 T^{9} + 6558458320 T^{10} + 47469543432 T^{11} + 335293827292 T^{12} + 47469543432 p T^{13} + 6558458320 p^{2} T^{14} + 968381160 p^{3} T^{15} + 128883791 p^{4} T^{16} + 14753484 p^{5} T^{17} + 1791912 p^{6} T^{18} + 225884 p^{7} T^{19} + 22578 p^{8} T^{20} + 44 p^{10} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
47 \( 1 + 122 p T^{4} + 19163791 T^{8} + 54284069110 T^{12} + 89055668533659 T^{16} + 97119285689140836 T^{20} + \)\(18\!\cdots\!94\)\( T^{24} + 97119285689140836 p^{4} T^{28} + 89055668533659 p^{8} T^{32} + 54284069110 p^{12} T^{36} + 19163791 p^{16} T^{40} + 122 p^{21} T^{44} + p^{24} T^{48} \)
53 \( 1 - 2876 T^{4} + 32919458 T^{8} - 106094932108 T^{12} + 504636826095503 T^{16} - 1620004442636812152 T^{20} + \)\(48\!\cdots\!04\)\( T^{24} - 1620004442636812152 p^{4} T^{28} + 504636826095503 p^{8} T^{32} - 106094932108 p^{12} T^{36} + 32919458 p^{16} T^{40} - 2876 p^{20} T^{44} + p^{24} T^{48} \)
59 \( ( 1 + 504 T^{2} + 122990 T^{4} + 19267192 T^{6} + 2165499663 T^{8} + 184637179600 T^{10} + 12280786220004 T^{12} + 184637179600 p^{2} T^{14} + 2165499663 p^{4} T^{16} + 19267192 p^{6} T^{18} + 122990 p^{8} T^{20} + 504 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
61 \( ( 1 - 8 T + 326 T^{2} - 2024 T^{3} + 45655 T^{4} - 223856 T^{5} + 3601364 T^{6} - 223856 p T^{7} + 45655 p^{2} T^{8} - 2024 p^{3} T^{9} + 326 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{4} \)
67 \( ( 1 - 12 T + 72 T^{2} - 364 T^{3} + 1762 T^{4} - 42036 T^{5} + 443816 T^{6} - 4321076 T^{7} + 425445 p T^{8} - 164037656 T^{9} + 1590485712 T^{10} - 19521072088 T^{11} + 228318456188 T^{12} - 19521072088 p T^{13} + 1590485712 p^{2} T^{14} - 164037656 p^{3} T^{15} + 425445 p^{5} T^{16} - 4321076 p^{5} T^{17} + 443816 p^{6} T^{18} - 42036 p^{7} T^{19} + 1762 p^{8} T^{20} - 364 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
71 \( ( 1 - 428 T^{2} + 95818 T^{4} - 14616604 T^{6} + 1695950527 T^{8} - 158323825976 T^{10} + 12274494567468 T^{12} - 158323825976 p^{2} T^{14} + 1695950527 p^{4} T^{16} - 14616604 p^{6} T^{18} + 95818 p^{8} T^{20} - 428 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
73 \( ( 1 - 16 T + 128 T^{2} - 816 T^{3} - 2058 T^{4} + 38288 T^{5} - 16256 T^{6} - 3006672 T^{7} + 71358319 T^{8} - 567676064 T^{9} + 3101927168 T^{10} - 14071432160 T^{11} + 13407621812 T^{12} - 14071432160 p T^{13} + 3101927168 p^{2} T^{14} - 567676064 p^{3} T^{15} + 71358319 p^{4} T^{16} - 3006672 p^{5} T^{17} - 16256 p^{6} T^{18} + 38288 p^{7} T^{19} - 2058 p^{8} T^{20} - 816 p^{9} T^{21} + 128 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
79 \( ( 1 - 354 T^{2} + 59143 T^{4} - 6202130 T^{6} + 445671739 T^{8} - 22229202188 T^{10} + 1161801775866 T^{12} - 22229202188 p^{2} T^{14} + 445671739 p^{4} T^{16} - 6202130 p^{6} T^{18} + 59143 p^{8} T^{20} - 354 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
83 \( 1 + 11180 T^{4} + 272303074 T^{8} + 2171408181692 T^{12} + 30248926653479119 T^{16} + \)\(18\!\cdots\!92\)\( T^{20} + \)\(18\!\cdots\!16\)\( T^{24} + \)\(18\!\cdots\!92\)\( p^{4} T^{28} + 30248926653479119 p^{8} T^{32} + 2171408181692 p^{12} T^{36} + 272303074 p^{16} T^{40} + 11180 p^{20} T^{44} + p^{24} T^{48} \)
89 \( ( 1 + 716 T^{2} + 251730 T^{4} + 57698908 T^{6} + 9611450511 T^{8} + 1225585609496 T^{10} + 122655863012092 T^{12} + 1225585609496 p^{2} T^{14} + 9611450511 p^{4} T^{16} + 57698908 p^{6} T^{18} + 251730 p^{8} T^{20} + 716 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
97 \( ( 1 - 12 T + 72 T^{2} - 8 T^{3} + 9339 T^{4} - 195304 T^{5} + 1671272 T^{6} - 4286972 T^{7} + 46362163 T^{8} - 1785089816 T^{9} + 21581114832 T^{10} - 87569792432 T^{11} - 300423376846 T^{12} - 87569792432 p T^{13} + 21581114832 p^{2} T^{14} - 1785089816 p^{3} T^{15} + 46362163 p^{4} T^{16} - 4286972 p^{5} T^{17} + 1671272 p^{6} T^{18} - 195304 p^{7} T^{19} + 9339 p^{8} T^{20} - 8 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.48264625043685039198652010157, −3.35834779303206182612879803259, −3.26506566590548787064319457337, −3.25865001457220604375637755349, −3.06180810509536809677146384878, −3.00840801999611030378276945856, −2.87468705437903145394906106039, −2.84756299872267457114284816361, −2.81380028136633989558056339741, −2.79725446955299344494612740399, −2.59320498130693397251481160305, −2.49308201053877021744170209761, −2.29182093906004056159753011241, −2.15500145594797824474335887863, −2.08071031822733052795428344790, −1.95593745688978063229320285775, −1.92125265782047009007551537095, −1.88209981289986690488224305437, −1.81373422795442416444879614727, −1.77910111275083157188588187375, −1.76119272322542106218804930515, −1.19412052019984611070403396522, −1.01286998318468232804770640116, −0.993962823957079229028518228624, −0.45003944395854031226311914675, 0.45003944395854031226311914675, 0.993962823957079229028518228624, 1.01286998318468232804770640116, 1.19412052019984611070403396522, 1.76119272322542106218804930515, 1.77910111275083157188588187375, 1.81373422795442416444879614727, 1.88209981289986690488224305437, 1.92125265782047009007551537095, 1.95593745688978063229320285775, 2.08071031822733052795428344790, 2.15500145594797824474335887863, 2.29182093906004056159753011241, 2.49308201053877021744170209761, 2.59320498130693397251481160305, 2.79725446955299344494612740399, 2.81380028136633989558056339741, 2.84756299872267457114284816361, 2.87468705437903145394906106039, 3.00840801999611030378276945856, 3.06180810509536809677146384878, 3.25865001457220604375637755349, 3.26506566590548787064319457337, 3.35834779303206182612879803259, 3.48264625043685039198652010157

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

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