Properties

Label 8-75e4-1.1-c9e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $2.22635\times 10^{6}$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.24e5·16-s − 6.76e6·31-s − 4.71e8·61-s − 3.87e8·81-s + 9.43e9·121-s + ⋯
L(s)  = 1  − 2·16-s − 1.31·31-s − 4.36·61-s − 81-s + 4·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.22635\times 10^{6}\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.3343220327\)
\(L(\frac12)\) \(\approx\) \(0.3343220327\)
\(L(\frac{11}{2})\) 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$C_2^2$ \( 1 + p^{18} T^{4} \)
5 \( 1 \)
good2$C_2$ \( ( 1 - p^{5} T + p^{9} T^{2} )^{2}( 1 + p^{5} T + p^{9} T^{2} )^{2} \)
7$C_2^3$ \( 1 + 2757049053441698 T^{4} + p^{36} T^{8} \)
11$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
13$C_2^3$ \( 1 - \)\(17\!\cdots\!42\)\( T^{4} + p^{36} T^{8} \)
17$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 308559680858 T^{2} + p^{18} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
31$C_2$ \( ( 1 + 1691228 T + p^{9} T^{2} )^{4} \)
37$C_2^3$ \( 1 - \)\(33\!\cdots\!42\)\( T^{4} + p^{36} T^{8} \)
41$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
43$C_2^3$ \( 1 + \)\(28\!\cdots\!98\)\( T^{4} + p^{36} T^{8} \)
47$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
61$C_2$ \( ( 1 + 117903058 T + p^{9} T^{2} )^{4} \)
67$C_2^3$ \( 1 + \)\(26\!\cdots\!18\)\( T^{4} + p^{36} T^{8} \)
71$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
73$C_2^3$ \( 1 - \)\(60\!\cdots\!62\)\( T^{4} + p^{36} T^{8} \)
79$C_2^2$ \( ( 1 + 140655567501204338 T^{2} + p^{18} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
97$C_2^3$ \( 1 - \)\(11\!\cdots\!22\)\( T^{4} + p^{36} T^{8} \)
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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

−8.837320862005306315340221882868, −8.675046737623734136584123105404, −8.246731677815366425356653935441, −7.84652813924331521823007899308, −7.56298565681549986283982865595, −7.37245134327516390497639798200, −6.78567365751891155390584756704, −6.78162504429410445762406286462, −6.45522241995490734489996448069, −5.81549055782276979553948310868, −5.74546662995937661657476214109, −5.40162171196117831722805179781, −4.80348974368373954967817238876, −4.53191575359585932695404139155, −4.27778403450699094046958501259, −4.10665209555466161875698577355, −3.20751323349692611058195441930, −3.14578044074516842061387578903, −2.90884596816198060266425717616, −2.05002086155482247184820147514, −1.87880342569820562167800176447, −1.79507032768832124632145415993, −0.995232416333677329534973895786, −0.59286205610586884259451132353, −0.093844874986505801546363170258, 0.093844874986505801546363170258, 0.59286205610586884259451132353, 0.995232416333677329534973895786, 1.79507032768832124632145415993, 1.87880342569820562167800176447, 2.05002086155482247184820147514, 2.90884596816198060266425717616, 3.14578044074516842061387578903, 3.20751323349692611058195441930, 4.10665209555466161875698577355, 4.27778403450699094046958501259, 4.53191575359585932695404139155, 4.80348974368373954967817238876, 5.40162171196117831722805179781, 5.74546662995937661657476214109, 5.81549055782276979553948310868, 6.45522241995490734489996448069, 6.78162504429410445762406286462, 6.78567365751891155390584756704, 7.37245134327516390497639798200, 7.56298565681549986283982865595, 7.84652813924331521823007899308, 8.246731677815366425356653935441, 8.675046737623734136584123105404, 8.837320862005306315340221882868

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