Properties

Label 4-4680e2-1.1-c1e2-0-1
Degree $4$
Conductor $21902400$
Sign $1$
Analytic cond. $1396.51$
Root an. cond. $6.11309$
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  + 2·5-s − 10·11-s + 8·19-s − 25-s − 8·29-s − 22·41-s − 11·49-s − 20·55-s + 24·59-s − 14·61-s + 14·71-s + 10·79-s − 6·89-s + 16·95-s + 12·101-s + 24·109-s + 53·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 0.894·5-s − 3.01·11-s + 1.83·19-s − 1/5·25-s − 1.48·29-s − 3.43·41-s − 1.57·49-s − 2.69·55-s + 3.12·59-s − 1.79·61-s + 1.66·71-s + 1.12·79-s − 0.635·89-s + 1.64·95-s + 1.19·101-s + 2.29·109-s + 4.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21902400\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1396.51\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21902400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6509842487\)
\(L(\frac12)\) \(\approx\) \(0.6509842487\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 193 T^{2} + p^{2} T^{4} \)
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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

−8.401311192729721344580035422274, −8.265269534654954141410990237911, −7.53552221236446321492870944747, −7.47015492295014851257914693387, −7.33765788628101262310077533277, −6.48103681477865901897981336596, −6.45410273965163771018212917258, −5.63204685621329517791850004295, −5.57922927235198182161478566706, −5.19159696553250854489100532840, −4.95080524062542653579973794126, −4.74262201340020394324254756137, −3.74406004926410892658320722383, −3.38409052496117103209751156006, −3.23825232365821563417356673251, −2.47766757880110020421314770561, −2.27551006542102180849151253786, −1.80757890890138271899321401355, −1.16649143083525911993073084617, −0.21861776551954550649651597637, 0.21861776551954550649651597637, 1.16649143083525911993073084617, 1.80757890890138271899321401355, 2.27551006542102180849151253786, 2.47766757880110020421314770561, 3.23825232365821563417356673251, 3.38409052496117103209751156006, 3.74406004926410892658320722383, 4.74262201340020394324254756137, 4.95080524062542653579973794126, 5.19159696553250854489100532840, 5.57922927235198182161478566706, 5.63204685621329517791850004295, 6.45410273965163771018212917258, 6.48103681477865901897981336596, 7.33765788628101262310077533277, 7.47015492295014851257914693387, 7.53552221236446321492870944747, 8.265269534654954141410990237911, 8.401311192729721344580035422274

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