Properties

Degree 8
Conductor $ 5^{4} $
Sign $1$
Motivic weight 9
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 340·4-s + 1.14e3·5-s + 4.51e4·9-s + 1.09e5·11-s + 2.90e5·16-s − 6.36e5·19-s + 3.87e5·20-s − 1.91e4·25-s − 3.53e6·29-s − 1.05e7·31-s + 1.53e7·36-s − 1.67e7·41-s + 3.73e7·44-s + 5.15e7·45-s + 5.72e7·49-s + 1.25e8·55-s − 4.60e8·59-s + 3.60e8·61-s + 2.47e8·64-s − 4.76e7·71-s − 2.16e8·76-s − 7.28e8·79-s + 3.31e8·80-s + 9.91e8·81-s − 1.58e9·89-s − 7.26e8·95-s + 4.96e9·99-s + ⋯
L(s)  = 1  + 0.664·4-s + 0.815·5-s + 2.29·9-s + 2.26·11-s + 1.10·16-s − 1.12·19-s + 0.541·20-s − 0.00980·25-s − 0.927·29-s − 2.05·31-s + 1.52·36-s − 0.927·41-s + 1.50·44-s + 1.87·45-s + 1.41·49-s + 1.84·55-s − 4.95·59-s + 3.33·61-s + 1.84·64-s − 0.222·71-s − 0.744·76-s − 2.10·79-s + 0.903·80-s + 2.56·81-s − 2.67·89-s − 0.914·95-s + 5.19·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(625\)    =    \(5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  induced by $\chi_{5} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)$
$L(5)$  $\approx$  $3.92542$
$L(\frac12)$  $\approx$  $3.92542$
$L(\frac{11}{2})$   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 8. If $p = 5$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad5$D_{4}$ \( 1 - 228 p T + 422 p^{5} T^{2} - 228 p^{10} T^{3} + p^{18} T^{4} \)
good2$C_2^2 \wr C_2$ \( 1 - 85 p^{2} T^{2} - 2733 p^{6} T^{4} - 85 p^{20} T^{6} + p^{36} T^{8} \)
3$C_2^2 \wr C_2$ \( 1 - 5020 p^{2} T^{2} + 12953638 p^{4} T^{4} - 5020 p^{20} T^{6} + p^{36} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 - 166900 p^{3} T^{2} + 1461117678198 p^{4} T^{4} - 166900 p^{21} T^{6} + p^{36} T^{8} \)
11$D_{4}$ \( ( 1 - 54984 T + 5180465446 T^{2} - 54984 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
13$C_2^2 \wr C_2$ \( 1 - 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} - 35613791860 p^{18} T^{6} + p^{36} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} - 285780369220 p^{18} T^{6} + p^{36} T^{8} \)
19$D_{4}$ \( ( 1 + 16760 p T + 505756418358 T^{2} + 16760 p^{10} T^{3} + p^{18} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - 5779790962540 p^{18} T^{6} + p^{36} T^{8} \)
29$D_{4}$ \( ( 1 + 1765860 T + 26950935551038 T^{2} + 1765860 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 5293856 T + 59464921598526 T^{2} + 5293856 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
37$C_2^2 \wr C_2$ \( 1 - 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} - 231603274936660 p^{18} T^{6} + p^{36} T^{8} \)
41$D_{4}$ \( ( 1 + 8394276 T + 221313076168966 T^{2} + 8394276 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 - 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} - 614109141147100 p^{18} T^{6} + p^{36} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 - 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - 1368976020813580 p^{18} T^{6} + p^{36} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} - 7684297973864980 p^{18} T^{6} + p^{36} T^{8} \)
59$D_{4}$ \( ( 1 + 230414520 T + 28555631923987078 T^{2} + 230414520 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 180245284 T + 30154717014478446 T^{2} - 180245284 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} + 41160407446058180 p^{18} T^{6} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 + 23805936 T + 85782754020107086 T^{2} + 23805936 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 - 229489314868712740 T^{2} + \)\(37\!\cdots\!22\)\( p^{2} T^{4} - 229489314868712740 p^{18} T^{6} + p^{36} T^{8} \)
79$D_{4}$ \( ( 1 + 364021760 T + 220545463862625438 T^{2} + 364021760 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - 434569632367965820 p^{18} T^{6} + p^{36} T^{8} \)
89$D_{4}$ \( ( 1 + 791350380 T + 832192702699668118 T^{2} + 791350380 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
97$C_2^2 \wr C_2$ \( 1 - 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - 2561123777205326980 p^{18} T^{6} + p^{36} T^{8} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.88656601889865950722610501348, −15.96632906290911725740572593786, −15.73086688659783430906865798029, −15.17597332297266355341544155098, −14.67760821266155244634966781811, −14.46261027238274674268282166665, −13.83244558470436788780220785247, −13.18334852706229502296228920541, −12.64232969756892670754402226057, −12.59051006570527268560620301989, −11.84098952926041106032905428276, −11.27038605898789196311788413999, −10.60380997660035783241402628780, −10.21196647938936155377652079279, −9.437150296272990796069135813970, −9.368755920511361841287164112360, −8.413826415017186276113723017656, −7.28687580313160429112295426927, −7.05083715819696954166285473446, −6.37816424446454197007858084399, −5.63865186807939055305227023641, −4.27132325936125202902816454889, −3.73921493109274457842441622083, −1.76511031061647646631543012965, −1.44920740204565842402946020490, 1.44920740204565842402946020490, 1.76511031061647646631543012965, 3.73921493109274457842441622083, 4.27132325936125202902816454889, 5.63865186807939055305227023641, 6.37816424446454197007858084399, 7.05083715819696954166285473446, 7.28687580313160429112295426927, 8.413826415017186276113723017656, 9.368755920511361841287164112360, 9.437150296272990796069135813970, 10.21196647938936155377652079279, 10.60380997660035783241402628780, 11.27038605898789196311788413999, 11.84098952926041106032905428276, 12.59051006570527268560620301989, 12.64232969756892670754402226057, 13.18334852706229502296228920541, 13.83244558470436788780220785247, 14.46261027238274674268282166665, 14.67760821266155244634966781811, 15.17597332297266355341544155098, 15.73086688659783430906865798029, 15.96632906290911725740572593786, 16.88656601889865950722610501348

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