Properties

Label 4-40e4-1.1-c1e2-0-8
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·9-s − 8·11-s + 8·19-s − 4·29-s − 16·31-s − 12·41-s − 2·49-s − 8·59-s + 4·61-s + 27·81-s + 12·89-s − 48·99-s − 12·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 48·171-s + 173-s + ⋯
L(s)  = 1  + 2·9-s − 2.41·11-s + 1.83·19-s − 0.742·29-s − 2.87·31-s − 1.87·41-s − 2/7·49-s − 1.04·59-s + 0.512·61-s + 3·81-s + 1.27·89-s − 4.82·99-s − 1.19·101-s + 2.68·109-s + 2.36·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 3.67·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630392283\)
\(L(\frac12)\) \(\approx\) \(1.630392283\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−9.572093029991495062249867714706, −9.270231862591860240656425884260, −9.017917827723684456430935223940, −8.140003332005757446198756522555, −8.017586346526220315568453851134, −7.39807223353356363562229469537, −7.36430741880210061266120750282, −7.10564226554235055524056984019, −6.43396732214031063076156644361, −5.83135606837318926877023171140, −5.33151221877542772152168056433, −5.03320658751984636300447285701, −4.92161818042559725828230006055, −4.06390688273939225570304188525, −3.62665358801877072391497326764, −3.24119016684682002831995332564, −2.61279107285671708396311829716, −1.79457675974558589641265003896, −1.63815028387202988186110531264, −0.48710251976651989983071855223, 0.48710251976651989983071855223, 1.63815028387202988186110531264, 1.79457675974558589641265003896, 2.61279107285671708396311829716, 3.24119016684682002831995332564, 3.62665358801877072391497326764, 4.06390688273939225570304188525, 4.92161818042559725828230006055, 5.03320658751984636300447285701, 5.33151221877542772152168056433, 5.83135606837318926877023171140, 6.43396732214031063076156644361, 7.10564226554235055524056984019, 7.36430741880210061266120750282, 7.39807223353356363562229469537, 8.017586346526220315568453851134, 8.140003332005757446198756522555, 9.017917827723684456430935223940, 9.270231862591860240656425884260, 9.572093029991495062249867714706

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