Properties

Label 4-45e4-1.1-c1e2-0-1
Degree $4$
Conductor $4100625$
Sign $1$
Analytic cond. $261.459$
Root an. cond. $4.02115$
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  + 4·4-s − 6·11-s + 12·16-s + 2·19-s + 18·29-s − 2·31-s + 6·41-s − 24·44-s + 10·49-s − 6·59-s − 20·61-s + 32·64-s − 6·71-s + 8·76-s + 32·79-s − 30·89-s + 18·101-s + 2·109-s + 72·116-s + 5·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s − 1.80·11-s + 3·16-s + 0.458·19-s + 3.34·29-s − 0.359·31-s + 0.937·41-s − 3.61·44-s + 10/7·49-s − 0.781·59-s − 2.56·61-s + 4·64-s − 0.712·71-s + 0.917·76-s + 3.60·79-s − 3.17·89-s + 1.79·101-s + 0.191·109-s + 6.68·116-s + 5/11·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4100625\)    =    \(3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(261.459\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4100625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.300791888\)
\(L(\frac12)\) \(\approx\) \(4.300791888\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−9.409265076593214826447202028670, −8.848148631199312682378781170908, −8.421563154696648728197810734967, −7.978920138078111384126912114818, −7.62706101882939484513659761584, −7.58984611297168225253388670802, −6.96213152393121783490717871446, −6.61909646144205238113422917343, −6.17209209093396426575799248006, −5.96480758774958513175120158572, −5.35579023265781539742934366489, −5.07420871472195861382190796192, −4.54310381076291900501773236270, −3.94095302669928419138325805927, −3.04401854990106258467557738750, −3.01658162898504592438145402597, −2.58147644182634272158888303380, −2.11370439186242015492247528047, −1.38758845020114474249724889051, −0.73804223298229897906345000775, 0.73804223298229897906345000775, 1.38758845020114474249724889051, 2.11370439186242015492247528047, 2.58147644182634272158888303380, 3.01658162898504592438145402597, 3.04401854990106258467557738750, 3.94095302669928419138325805927, 4.54310381076291900501773236270, 5.07420871472195861382190796192, 5.35579023265781539742934366489, 5.96480758774958513175120158572, 6.17209209093396426575799248006, 6.61909646144205238113422917343, 6.96213152393121783490717871446, 7.58984611297168225253388670802, 7.62706101882939484513659761584, 7.978920138078111384126912114818, 8.421563154696648728197810734967, 8.848148631199312682378781170908, 9.409265076593214826447202028670

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