Properties

Label 4-1148e2-1.1-c1e2-0-2
Degree $4$
Conductor $1317904$
Sign $1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
Motivic weight $1$
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-s + 3·5-s + 5·7-s + 3·9-s − 3·11-s − 8·13-s − 3·15-s + 7·19-s − 5·21-s − 6·23-s + 5·25-s − 8·27-s + 12·29-s + 10·31-s + 3·33-s + 15·35-s − 2·37-s + 8·39-s − 2·41-s − 8·43-s + 9·45-s − 12·47-s + 18·49-s + 6·53-s − 9·55-s − 7·57-s − 6·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 1.88·7-s + 9-s − 0.904·11-s − 2.21·13-s − 0.774·15-s + 1.60·19-s − 1.09·21-s − 1.25·23-s + 25-s − 1.53·27-s + 2.22·29-s + 1.79·31-s + 0.522·33-s + 2.53·35-s − 0.328·37-s + 1.28·39-s − 0.312·41-s − 1.21·43-s + 1.34·45-s − 1.75·47-s + 18/7·49-s + 0.824·53-s − 1.21·55-s − 0.927·57-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1148} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.802088179\)
\(L(\frac12)\) \(\approx\) \(2.802088179\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
41$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.19106632542635086340634989061, −9.745729731948265619225177278201, −9.480014696826783134729229895965, −8.698030150360916937617584052215, −8.270633291709258339055474741293, −7.87167810182071626439439879439, −7.63486063245304789731337065657, −7.06770172590634087964419016770, −6.78910354742494506476077676976, −6.07925906365315351098141303553, −5.73073252471695500013273618058, −5.00807182546260997785011896876, −5.00204516769489710156568694167, −4.76953954725935704611228466520, −4.16636081779854867182988028584, −3.10841682492313998810225076496, −2.59876736114822818325043109856, −1.98504048598254542901029208621, −1.65635820597520950498015914372, −0.77575176012664874582822881449, 0.77575176012664874582822881449, 1.65635820597520950498015914372, 1.98504048598254542901029208621, 2.59876736114822818325043109856, 3.10841682492313998810225076496, 4.16636081779854867182988028584, 4.76953954725935704611228466520, 5.00204516769489710156568694167, 5.00807182546260997785011896876, 5.73073252471695500013273618058, 6.07925906365315351098141303553, 6.78910354742494506476077676976, 7.06770172590634087964419016770, 7.63486063245304789731337065657, 7.87167810182071626439439879439, 8.270633291709258339055474741293, 8.698030150360916937617584052215, 9.480014696826783134729229895965, 9.745729731948265619225177278201, 10.19106632542635086340634989061

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