Properties

Label 4-1840e2-1.1-c1e2-0-5
Degree $4$
Conductor $3385600$
Sign $1$
Analytic cond. $215.868$
Root an. cond. $3.83307$
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  + 4·5-s + 2·9-s + 16·19-s + 11·25-s + 10·29-s + 10·31-s − 14·41-s + 8·45-s + 13·49-s + 6·59-s − 12·61-s − 26·71-s − 28·79-s − 5·81-s + 28·89-s + 64·95-s + 30·101-s − 36·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.78·5-s + 2/3·9-s + 3.67·19-s + 11/5·25-s + 1.85·29-s + 1.79·31-s − 2.18·41-s + 1.19·45-s + 13/7·49-s + 0.781·59-s − 1.53·61-s − 3.08·71-s − 3.15·79-s − 5/9·81-s + 2.96·89-s + 6.56·95-s + 2.98·101-s − 3.44·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3385600\)    =    \(2^{8} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(215.868\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3385600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.163173229\)
\(L(\frac12)\) \(\approx\) \(5.163173229\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.576148030410108098600644602424, −8.951532487729763002332319976483, −8.937412046044829089611044227641, −8.375765579340527809822076059588, −7.76313551498356507799038903369, −7.43975023962131211940246693187, −7.00935219860697231240192946526, −6.74325802401862703672458195401, −6.09269721317024906131007890852, −5.95137489070646669466531836892, −5.33328016647002968014205843757, −5.09448648118850537508863782361, −4.70235207395148745835275342817, −4.18722899653004125931631119656, −3.32978838267781671944453397696, −2.84274349368173337774974213154, −2.81783495171724879901785858667, −1.83492881296117993969960863091, −1.17306808228243964742902952309, −1.07977648410735711665803284198, 1.07977648410735711665803284198, 1.17306808228243964742902952309, 1.83492881296117993969960863091, 2.81783495171724879901785858667, 2.84274349368173337774974213154, 3.32978838267781671944453397696, 4.18722899653004125931631119656, 4.70235207395148745835275342817, 5.09448648118850537508863782361, 5.33328016647002968014205843757, 5.95137489070646669466531836892, 6.09269721317024906131007890852, 6.74325802401862703672458195401, 7.00935219860697231240192946526, 7.43975023962131211940246693187, 7.76313551498356507799038903369, 8.375765579340527809822076059588, 8.937412046044829089611044227641, 8.951532487729763002332319976483, 9.576148030410108098600644602424

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