Properties

Label 8-75e4-1.1-c7e4-0-3
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $301304.$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.27e4·16-s + 1.32e6·31-s + 7.99e6·61-s − 4.78e6·81-s + 7.79e7·121-s + ⋯
L(s)  = 1  − 2·16-s + 7.99·31-s + 4.50·61-s − 81-s + 4·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(301304.\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :7/2, 7/2, 7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(6.289773048\)
\(L(\frac12)\) \(\approx\) \(6.289773048\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2^2$ \( 1 + p^{14} T^{4} \)
5 \( 1 \)
good2$C_2$ \( ( 1 - p^{4} T + p^{7} T^{2} )^{2}( 1 + p^{4} T + p^{7} T^{2} )^{2} \)
7$C_2^3$ \( 1 - 1351253357977 T^{4} + p^{28} T^{8} \)
11$C_2$ \( ( 1 - p^{7} T^{2} )^{4} \)
13$C_2^3$ \( 1 - 6759928051400497 T^{4} + p^{28} T^{8} \)
17$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 69090803 T^{2} + p^{14} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p^{7} T^{2} )^{4} \)
31$C_2$ \( ( 1 - 331387 T + p^{7} T^{2} )^{4} \)
37$C_2^3$ \( 1 - \)\(55\!\cdots\!22\)\( T^{4} + p^{28} T^{8} \)
41$C_2$ \( ( 1 - p^{7} T^{2} )^{4} \)
43$C_2^3$ \( 1 - \)\(64\!\cdots\!77\)\( T^{4} + p^{28} T^{8} \)
47$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p^{7} T^{2} )^{4} \)
61$C_2$ \( ( 1 - 1998347 T + p^{7} T^{2} )^{4} \)
67$C_2^3$ \( 1 - \)\(54\!\cdots\!17\)\( T^{4} + p^{28} T^{8} \)
71$C_2$ \( ( 1 - p^{7} T^{2} )^{4} \)
73$C_2^3$ \( 1 + \)\(54\!\cdots\!18\)\( T^{4} + p^{28} T^{8} \)
79$C_2^2$ \( ( 1 + 38383122173618 T^{2} + p^{14} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{14} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p^{7} T^{2} )^{4} \)
97$C_2^3$ \( 1 + \)\(80\!\cdots\!63\)\( T^{4} + p^{28} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.597897021881650634476510055251, −8.690745060066492999943298834893, −8.511780635309880095552932667015, −8.372367819203288381806323777071, −8.355823144415181067062732741757, −7.87085441230304205755861889408, −7.15533360002861009485123078021, −7.11778032868994693899629600570, −6.69404845482672026131120249335, −6.44593989692216236574455233093, −6.17331797534105332116258953721, −5.90810090013092891687380781698, −5.20759614301632545135376167470, −4.89510612831368179892121918956, −4.54431658249328599439511447028, −4.37438941127491511963557531483, −4.09314184501363984964519300960, −3.34153123051256104667015057056, −2.88333098512087756508732556491, −2.48081779188899308855263052700, −2.45209306092126336171779485861, −1.76692303734727413390453388789, −0.957809599224333471992722726302, −0.76305423110524817856720216020, −0.51510783714260050662429927666, 0.51510783714260050662429927666, 0.76305423110524817856720216020, 0.957809599224333471992722726302, 1.76692303734727413390453388789, 2.45209306092126336171779485861, 2.48081779188899308855263052700, 2.88333098512087756508732556491, 3.34153123051256104667015057056, 4.09314184501363984964519300960, 4.37438941127491511963557531483, 4.54431658249328599439511447028, 4.89510612831368179892121918956, 5.20759614301632545135376167470, 5.90810090013092891687380781698, 6.17331797534105332116258953721, 6.44593989692216236574455233093, 6.69404845482672026131120249335, 7.11778032868994693899629600570, 7.15533360002861009485123078021, 7.87085441230304205755861889408, 8.355823144415181067062732741757, 8.372367819203288381806323777071, 8.511780635309880095552932667015, 8.690745060066492999943298834893, 9.597897021881650634476510055251

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