Properties

Degree 16
Conductor $ 5^{8} $
Sign $1$
Motivic weight 17
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.34e5·4-s + 3.79e5·5-s + 3.99e8·9-s + 4.63e8·11-s + 1.62e10·16-s − 2.06e10·19-s + 8.90e10·20-s − 8.22e11·25-s − 4.07e12·29-s + 1.13e13·31-s + 9.36e13·36-s + 9.72e13·41-s + 1.08e14·44-s + 1.51e14·45-s + 5.02e14·49-s + 1.75e14·55-s − 1.09e15·59-s + 8.06e15·61-s − 4.17e14·64-s + 2.52e16·71-s − 4.83e15·76-s + 3.22e15·79-s + 6.17e15·80-s + 5.97e16·81-s + 9.29e16·89-s − 7.81e15·95-s + 1.84e17·99-s + ⋯
L(s)  = 1  + 1.79·4-s + 0.434·5-s + 3.09·9-s + 0.651·11-s + 0.948·16-s − 0.278·19-s + 0.777·20-s − 1.07·25-s − 1.51·29-s + 2.38·31-s + 5.53·36-s + 1.90·41-s + 1.16·44-s + 1.34·45-s + 2.15·49-s + 0.282·55-s − 0.967·59-s + 5.38·61-s − 0.185·64-s + 4.63·71-s − 0.498·76-s + 0.239·79-s + 0.411·80-s + 3.58·81-s + 2.50·89-s − 0.120·95-s + 2.01·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+17/2)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(390625\)    =    \(5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(17\)
character  :  induced by $\chi_{5} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 390625,\ (\ :[17/2]^{8}),\ 1)$
$L(9)$  $\approx$  $35.4432$
$L(\frac12)$  $\approx$  $35.4432$
$L(\frac{19}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 16. If $p = 5$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad5 \( 1 - 15168 p^{2} T + 309286732 p^{5} T^{2} + 177748146432 p^{9} T^{3} + 2397588578934 p^{16} T^{4} + 177748146432 p^{26} T^{5} + 309286732 p^{39} T^{6} - 15168 p^{53} T^{7} + p^{68} T^{8} \)
good2 \( 1 - 58685 p^{2} T^{2} + 606488299 p^{6} T^{4} - 148653192985 p^{15} T^{6} + 178370738447659 p^{22} T^{8} - 148653192985 p^{49} T^{10} + 606488299 p^{74} T^{12} - 58685 p^{104} T^{14} + p^{136} T^{16} \)
3 \( 1 - 44344720 p^{2} T^{2} + 45525406383748 p^{7} T^{4} - \)\(30\!\cdots\!40\)\( p^{10} T^{6} + \)\(65\!\cdots\!94\)\( p^{18} T^{8} - \)\(30\!\cdots\!40\)\( p^{44} T^{10} + 45525406383748 p^{75} T^{12} - 44344720 p^{104} T^{14} + p^{136} T^{16} \)
7 \( 1 - 502234877138000 T^{2} + \)\(38\!\cdots\!04\)\( p^{2} T^{4} - \)\(75\!\cdots\!00\)\( p^{7} T^{6} + \)\(57\!\cdots\!94\)\( p^{10} T^{8} - \)\(75\!\cdots\!00\)\( p^{41} T^{10} + \)\(38\!\cdots\!04\)\( p^{70} T^{12} - 502234877138000 p^{102} T^{14} + p^{136} T^{16} \)
11 \( ( 1 - 231648288 T + 64273843738603108 p T^{2} + \)\(16\!\cdots\!84\)\( p^{2} T^{3} + \)\(34\!\cdots\!70\)\( p^{3} T^{4} + \)\(16\!\cdots\!84\)\( p^{19} T^{5} + 64273843738603108 p^{35} T^{6} - 231648288 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
13 \( 1 - 39268035547036882760 T^{2} + \)\(47\!\cdots\!24\)\( p^{2} T^{4} - \)\(39\!\cdots\!20\)\( p^{4} T^{6} + \)\(23\!\cdots\!14\)\( p^{6} T^{8} - \)\(39\!\cdots\!20\)\( p^{38} T^{10} + \)\(47\!\cdots\!24\)\( p^{70} T^{12} - 39268035547036882760 p^{102} T^{14} + p^{136} T^{16} \)
17 \( 1 - \)\(25\!\cdots\!20\)\( T^{2} + \)\(32\!\cdots\!16\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{6} + \)\(29\!\cdots\!46\)\( T^{8} - \)\(32\!\cdots\!40\)\( p^{34} T^{10} + \)\(32\!\cdots\!16\)\( p^{68} T^{12} - \)\(25\!\cdots\!20\)\( p^{102} T^{14} + p^{136} T^{16} \)
19 \( ( 1 + 10307856640 T + \)\(16\!\cdots\!56\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!26\)\( T^{4} + \)\(12\!\cdots\!80\)\( p^{17} T^{5} + \)\(16\!\cdots\!56\)\( p^{34} T^{6} + 10307856640 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
23 \( 1 - \)\(60\!\cdots\!40\)\( T^{2} + \)\(19\!\cdots\!36\)\( T^{4} - \)\(43\!\cdots\!80\)\( T^{6} + \)\(71\!\cdots\!86\)\( T^{8} - \)\(43\!\cdots\!80\)\( p^{34} T^{10} + \)\(19\!\cdots\!36\)\( p^{68} T^{12} - \)\(60\!\cdots\!40\)\( p^{102} T^{14} + p^{136} T^{16} \)
29 \( ( 1 + 2039508912360 T + \)\(21\!\cdots\!36\)\( T^{2} + \)\(45\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!86\)\( T^{4} + \)\(45\!\cdots\!20\)\( p^{17} T^{5} + \)\(21\!\cdots\!36\)\( p^{34} T^{6} + 2039508912360 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
31 \( ( 1 - 5664664329248 T + \)\(88\!\cdots\!08\)\( T^{2} - \)\(32\!\cdots\!96\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(32\!\cdots\!96\)\( p^{17} T^{5} + \)\(88\!\cdots\!08\)\( p^{34} T^{6} - 5664664329248 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
37 \( 1 - \)\(27\!\cdots\!60\)\( T^{2} + \)\(34\!\cdots\!56\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!26\)\( T^{8} - \)\(26\!\cdots\!20\)\( p^{34} T^{10} + \)\(34\!\cdots\!56\)\( p^{68} T^{12} - \)\(27\!\cdots\!60\)\( p^{102} T^{14} + p^{136} T^{16} \)
41 \( ( 1 - 48608626423728 T + \)\(55\!\cdots\!68\)\( T^{2} - \)\(40\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(40\!\cdots\!76\)\( p^{17} T^{5} + \)\(55\!\cdots\!68\)\( p^{34} T^{6} - 48608626423728 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
43 \( 1 - \)\(31\!\cdots\!00\)\( T^{2} + \)\(49\!\cdots\!96\)\( T^{4} - \)\(50\!\cdots\!00\)\( T^{6} + \)\(34\!\cdots\!06\)\( T^{8} - \)\(50\!\cdots\!00\)\( p^{34} T^{10} + \)\(49\!\cdots\!96\)\( p^{68} T^{12} - \)\(31\!\cdots\!00\)\( p^{102} T^{14} + p^{136} T^{16} \)
47 \( 1 - \)\(12\!\cdots\!80\)\( T^{2} + \)\(79\!\cdots\!76\)\( T^{4} - \)\(32\!\cdots\!60\)\( T^{6} + \)\(97\!\cdots\!66\)\( T^{8} - \)\(32\!\cdots\!60\)\( p^{34} T^{10} + \)\(79\!\cdots\!76\)\( p^{68} T^{12} - \)\(12\!\cdots\!80\)\( p^{102} T^{14} + p^{136} T^{16} \)
53 \( 1 - \)\(11\!\cdots\!80\)\( T^{2} + \)\(59\!\cdots\!76\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{6} + \)\(47\!\cdots\!66\)\( T^{8} - \)\(19\!\cdots\!60\)\( p^{34} T^{10} + \)\(59\!\cdots\!76\)\( p^{68} T^{12} - \)\(11\!\cdots\!80\)\( p^{102} T^{14} + p^{136} T^{16} \)
59 \( ( 1 + 545704756120320 T + \)\(31\!\cdots\!76\)\( T^{2} + \)\(93\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!66\)\( T^{4} + \)\(93\!\cdots\!40\)\( p^{17} T^{5} + \)\(31\!\cdots\!76\)\( p^{34} T^{6} + 545704756120320 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
61 \( ( 1 - 4032365505387488 T + \)\(10\!\cdots\!88\)\( T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!70\)\( T^{4} - \)\(18\!\cdots\!36\)\( p^{17} T^{5} + \)\(10\!\cdots\!88\)\( p^{34} T^{6} - 4032365505387488 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
67 \( 1 - \)\(56\!\cdots\!20\)\( T^{2} + \)\(16\!\cdots\!16\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!46\)\( T^{8} - \)\(30\!\cdots\!40\)\( p^{34} T^{10} + \)\(16\!\cdots\!16\)\( p^{68} T^{12} - \)\(56\!\cdots\!20\)\( p^{102} T^{14} + p^{136} T^{16} \)
71 \( ( 1 - 12620746029070368 T + \)\(16\!\cdots\!48\)\( T^{2} - \)\(11\!\cdots\!16\)\( T^{3} + \)\(81\!\cdots\!70\)\( T^{4} - \)\(11\!\cdots\!16\)\( p^{17} T^{5} + \)\(16\!\cdots\!48\)\( p^{34} T^{6} - 12620746029070368 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
73 \( 1 - \)\(25\!\cdots\!40\)\( T^{2} + \)\(30\!\cdots\!36\)\( T^{4} - \)\(24\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!86\)\( T^{8} - \)\(24\!\cdots\!80\)\( p^{34} T^{10} + \)\(30\!\cdots\!36\)\( p^{68} T^{12} - \)\(25\!\cdots\!40\)\( p^{102} T^{14} + p^{136} T^{16} \)
79 \( ( 1 - 20442103403360 p T + \)\(57\!\cdots\!36\)\( T^{2} - \)\(80\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(80\!\cdots\!80\)\( p^{17} T^{5} + \)\(57\!\cdots\!36\)\( p^{34} T^{6} - 20442103403360 p^{52} T^{7} + p^{68} T^{8} )^{2} \)
83 \( 1 - \)\(13\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!52\)\( p T^{4} - \)\(79\!\cdots\!40\)\( T^{6} + \)\(37\!\cdots\!46\)\( T^{8} - \)\(79\!\cdots\!40\)\( p^{34} T^{10} + \)\(14\!\cdots\!52\)\( p^{69} T^{12} - \)\(13\!\cdots\!20\)\( p^{102} T^{14} + p^{136} T^{16} \)
89 \( ( 1 - 46481993767813320 T + \)\(39\!\cdots\!16\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(70\!\cdots\!46\)\( T^{4} - \)\(17\!\cdots\!40\)\( p^{17} T^{5} + \)\(39\!\cdots\!16\)\( p^{34} T^{6} - 46481993767813320 p^{51} T^{7} + p^{68} T^{8} )^{2} \)
97 \( 1 - \)\(28\!\cdots\!80\)\( T^{2} + \)\(41\!\cdots\!76\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{6} + \)\(26\!\cdots\!66\)\( T^{8} - \)\(38\!\cdots\!60\)\( p^{34} T^{10} + \)\(41\!\cdots\!76\)\( p^{68} T^{12} - \)\(28\!\cdots\!80\)\( p^{102} T^{14} + p^{136} T^{16} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.85501291042625067932405510476, −7.35592137097598665093700409895, −7.21136442477289125728459447530, −6.83204213793252780556606918871, −6.79752005992018446053471276608, −6.77381539174281470743275851829, −6.17580734080903143768899000099, −5.89468869040389866869857399242, −5.75839417192414805430177023377, −5.24891551274259288026082385505, −4.77557233054311840163398467213, −4.61312861597637042206629989323, −4.16927888261452006715223820268, −3.83868910019167239092593741020, −3.80901103716293605637354387970, −3.44869972202378375392138932058, −2.69952514967163291414011894809, −2.43200733452414992725343338187, −2.02167465799960000664627427316, −2.00376158438617532957011914310, −1.94433282333420287482938492760, −1.11127694271998320005496516243, −0.944887904975419082232510492960, −0.894398863380974756150534442666, −0.45987989328136317256592664141, 0.45987989328136317256592664141, 0.894398863380974756150534442666, 0.944887904975419082232510492960, 1.11127694271998320005496516243, 1.94433282333420287482938492760, 2.00376158438617532957011914310, 2.02167465799960000664627427316, 2.43200733452414992725343338187, 2.69952514967163291414011894809, 3.44869972202378375392138932058, 3.80901103716293605637354387970, 3.83868910019167239092593741020, 4.16927888261452006715223820268, 4.61312861597637042206629989323, 4.77557233054311840163398467213, 5.24891551274259288026082385505, 5.75839417192414805430177023377, 5.89468869040389866869857399242, 6.17580734080903143768899000099, 6.77381539174281470743275851829, 6.79752005992018446053471276608, 6.83204213793252780556606918871, 7.21136442477289125728459447530, 7.35592137097598665093700409895, 7.85501291042625067932405510476

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

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