Properties

Label 4-2550e2-1.1-c1e2-0-30
Degree $4$
Conductor $6502500$
Sign $1$
Analytic cond. $414.605$
Root an. cond. $4.51241$
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-s − 9-s + 16-s + 8·19-s − 20·31-s + 36-s + 12·41-s + 10·49-s + 24·59-s + 16·61-s − 64-s + 12·71-s − 8·76-s + 20·79-s + 81-s + 36·89-s + 12·101-s − 40·109-s − 22·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s + 1.83·19-s − 3.59·31-s + 1/6·36-s + 1.87·41-s + 10/7·49-s + 3.12·59-s + 2.04·61-s − 1/8·64-s + 1.42·71-s − 0.917·76-s + 2.25·79-s + 1/9·81-s + 3.81·89-s + 1.19·101-s − 3.83·109-s − 2·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6502500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(414.605\)
Root analytic conductor: \(4.51241\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6502500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.281541349\)
\(L(\frac12)\) \(\approx\) \(2.281541349\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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

−9.095932902205181179578372046337, −8.972896758696052883051112482108, −8.193278888294562177613671787076, −7.993753500361472324876948423776, −7.56872729140441273204203570227, −7.26876052801828573588868265475, −6.84540714943143656177757857363, −6.48479365108209866322355732707, −5.73925305197396526310405932956, −5.54119792924237596299815788548, −5.14965293789467323514001022207, −5.09749030201889466224237422497, −4.08138226033058356006880172201, −3.72978536930896817466808366790, −3.70871785708933881972579842798, −2.97257568923183612670383180011, −2.25220788030208594698013464688, −2.05629181597688770640909017754, −0.991680808067481084250114185412, −0.62674347670875099260631967967, 0.62674347670875099260631967967, 0.991680808067481084250114185412, 2.05629181597688770640909017754, 2.25220788030208594698013464688, 2.97257568923183612670383180011, 3.70871785708933881972579842798, 3.72978536930896817466808366790, 4.08138226033058356006880172201, 5.09749030201889466224237422497, 5.14965293789467323514001022207, 5.54119792924237596299815788548, 5.73925305197396526310405932956, 6.48479365108209866322355732707, 6.84540714943143656177757857363, 7.26876052801828573588868265475, 7.56872729140441273204203570227, 7.993753500361472324876948423776, 8.193278888294562177613671787076, 8.972896758696052883051112482108, 9.095932902205181179578372046337

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