Properties

Label 4-6026e2-1.1-c1e2-0-0
Degree $4$
Conductor $36312676$
Sign $1$
Analytic cond. $2315.32$
Root an. cond. $6.93670$
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  + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 4·7-s + 4·8-s + 6·12-s − 2·13-s + 8·14-s + 5·16-s + 10·19-s + 8·21-s − 2·23-s + 8·24-s − 7·25-s − 4·26-s − 2·27-s + 12·28-s − 12·29-s + 16·31-s + 6·32-s − 8·37-s + 20·38-s − 4·39-s + 6·41-s + 16·42-s − 8·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.51·7-s + 1.41·8-s + 1.73·12-s − 0.554·13-s + 2.13·14-s + 5/4·16-s + 2.29·19-s + 1.74·21-s − 0.417·23-s + 1.63·24-s − 7/5·25-s − 0.784·26-s − 0.384·27-s + 2.26·28-s − 2.22·29-s + 2.87·31-s + 1.06·32-s − 1.31·37-s + 3.24·38-s − 0.640·39-s + 0.937·41-s + 2.46·42-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36312676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36312676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36312676\)    =    \(2^{2} \cdot 23^{2} \cdot 131^{2}\)
Sign: $1$
Analytic conductor: \(2315.32\)
Root analytic conductor: \(6.93670\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36312676,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(16.41261222\)
\(L(\frac12)\) \(\approx\) \(16.41261222\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
131$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 99 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 18 T + 220 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 120 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 198 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.091786284003627459929145465648, −7.933506902236091794358774228026, −7.45413960591466768163952019674, −7.40437819120540000938462799685, −6.78633569315099270772674198273, −6.57852349605830215193698653946, −5.85332478058135211891524915396, −5.58798529720733846884818179485, −5.35934681291178715900832323082, −5.06586478027751163261351894112, −4.53128499341987000471174941475, −4.32903961427680766092909003732, −3.64607699838655699789428732348, −3.60038814476370949173414212176, −3.02714417335520480234224273755, −2.71165805347071475712866635855, −2.13267720055965556463033352359, −1.95291978279713526175550188647, −1.39238338689426355639867609642, −0.70871907344883805469090508383, 0.70871907344883805469090508383, 1.39238338689426355639867609642, 1.95291978279713526175550188647, 2.13267720055965556463033352359, 2.71165805347071475712866635855, 3.02714417335520480234224273755, 3.60038814476370949173414212176, 3.64607699838655699789428732348, 4.32903961427680766092909003732, 4.53128499341987000471174941475, 5.06586478027751163261351894112, 5.35934681291178715900832323082, 5.58798529720733846884818179485, 5.85332478058135211891524915396, 6.57852349605830215193698653946, 6.78633569315099270772674198273, 7.40437819120540000938462799685, 7.45413960591466768163952019674, 7.933506902236091794358774228026, 8.091786284003627459929145465648

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