Properties

Label 4-5040e2-1.1-c1e2-0-0
Degree $4$
Conductor $25401600$
Sign $1$
Analytic cond. $1619.62$
Root an. cond. $6.34386$
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 − 4·11-s − 4·19-s − 25-s − 12·29-s − 12·31-s − 12·41-s − 49-s − 8·55-s + 16·59-s + 20·61-s − 28·71-s + 8·79-s + 20·89-s − 8·95-s + 20·101-s + 12·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s − 24·155-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s − 0.917·19-s − 1/5·25-s − 2.22·29-s − 2.15·31-s − 1.87·41-s − 1/7·49-s − 1.07·55-s + 2.08·59-s + 2.56·61-s − 3.32·71-s + 0.900·79-s + 2.11·89-s − 0.820·95-s + 1.99·101-s + 1.14·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 1.92·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25401600\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1619.62\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5040} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25401600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2911278422\)
\(L(\frac12)\) \(\approx\) \(0.2911278422\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.721615533959288411450524021992, −7.86202412993091424911309139482, −7.78637554285595287807837674701, −7.23982752453284554159702095117, −7.18266884366156334835730297336, −6.44543625275410000622458986563, −6.34939921314195853668822334234, −5.79806467560775040801618813002, −5.38835197531502580602532128080, −5.22499556027697807107143909652, −5.06548870694745889855017109045, −4.21799697076741159139333303547, −3.93061120079318067017096599878, −3.50282581088394799723528276198, −3.16732031872953320590923270595, −2.31092564693398283615615400929, −2.17642524197737091710857923314, −1.88867449626509289012333985965, −1.19493067099303979779355492198, −0.13876351859999194898018041976, 0.13876351859999194898018041976, 1.19493067099303979779355492198, 1.88867449626509289012333985965, 2.17642524197737091710857923314, 2.31092564693398283615615400929, 3.16732031872953320590923270595, 3.50282581088394799723528276198, 3.93061120079318067017096599878, 4.21799697076741159139333303547, 5.06548870694745889855017109045, 5.22499556027697807107143909652, 5.38835197531502580602532128080, 5.79806467560775040801618813002, 6.34939921314195853668822334234, 6.44543625275410000622458986563, 7.18266884366156334835730297336, 7.23982752453284554159702095117, 7.78637554285595287807837674701, 7.86202412993091424911309139482, 8.721615533959288411450524021992

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