Properties

Degree 8
Conductor $ 5^{8} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·11-s + 7·16-s − 152·31-s + 228·41-s − 112·61-s + 168·71-s − 63·81-s − 192·101-s − 394·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 84·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 1.09·11-s + 7/16·16-s − 4.90·31-s + 5.56·41-s − 1.83·61-s + 2.36·71-s − 7/9·81-s − 1.90·101-s − 3.25·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s − 0.477·176-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(390625\)    =    \(5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{25} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 390625,\ (\ :1, 1, 1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.711859$
$L(\frac12)$  $\approx$  $0.711859$
$L(2)$   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 \( 1 \)
good2$C_2^3$ \( 1 - 7 T^{4} + p^{8} T^{8} \)
3$C_2^3$ \( 1 + 7 p^{2} T^{4} + p^{8} T^{8} \)
7$C_2^3$ \( 1 - 2302 T^{4} + p^{8} T^{8} \)
11$C_2$ \( ( 1 + 3 T + p^{2} T^{2} )^{4} \)
13$C_2^3$ \( 1 - 4222 T^{4} + p^{8} T^{8} \)
17$C_2^3$ \( 1 - 120817 T^{4} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 697 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 338782 T^{4} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 - 782 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 38 T + p^{2} T^{2} )^{4} \)
37$C_2^3$ \( 1 + 132578 T^{4} + p^{8} T^{8} \)
41$C_2$ \( ( 1 - 57 T + p^{2} T^{2} )^{4} \)
43$C_2^3$ \( 1 + 6484898 T^{4} + p^{8} T^{8} \)
47$C_2^3$ \( 1 + 8816738 T^{4} + p^{8} T^{8} \)
53$C_2^3$ \( 1 - 2892862 T^{4} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 + 1138 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 28 T + p^{2} T^{2} )^{4} \)
67$C_2^3$ \( 1 - 20810017 T^{4} + p^{8} T^{8} \)
71$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )^{4} \)
73$C_2^3$ \( 1 + 49190543 T^{4} + p^{8} T^{8} \)
79$C_2^2$ \( ( 1 - 6082 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 27517583 T^{4} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 - 15617 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^3$ \( 1 - 7798462 T^{4} + p^{8} T^{8} \)
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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

−12.92108624713706615057147601345, −12.77679944275163605300715426036, −12.57910233808340025768232932523, −12.39871966517840062865009927183, −11.72130267965928507532906959660, −11.20975637095062978438764001943, −10.87036483367697919382493644174, −10.80202217950719097990488636664, −10.65256669380755182035270122951, −9.843700278017067399731181305591, −9.354066587400336108683727151781, −9.231979411563966906372676961873, −9.075493128417653782841684094869, −8.181050810439558547923718676808, −7.902577928817560403427339720800, −7.43070867433979855000875174944, −7.36806039937676854566796043849, −6.66659017825642368793563998673, −5.90391852040235819298130691681, −5.50779105914296339937998810838, −5.45386990868911896353903542376, −4.38168152664509159724498301141, −3.96617731422108493669297013406, −3.11856284495437891957650319790, −2.18276126975121981967295505990, 2.18276126975121981967295505990, 3.11856284495437891957650319790, 3.96617731422108493669297013406, 4.38168152664509159724498301141, 5.45386990868911896353903542376, 5.50779105914296339937998810838, 5.90391852040235819298130691681, 6.66659017825642368793563998673, 7.36806039937676854566796043849, 7.43070867433979855000875174944, 7.902577928817560403427339720800, 8.181050810439558547923718676808, 9.075493128417653782841684094869, 9.231979411563966906372676961873, 9.354066587400336108683727151781, 9.843700278017067399731181305591, 10.65256669380755182035270122951, 10.80202217950719097990488636664, 10.87036483367697919382493644174, 11.20975637095062978438764001943, 11.72130267965928507532906959660, 12.39871966517840062865009927183, 12.57910233808340025768232932523, 12.77679944275163605300715426036, 12.92108624713706615057147601345

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