Properties

Label 4-5070e2-1.1-c1e2-0-7
Degree $4$
Conductor $25704900$
Sign $1$
Analytic cond. $1638.96$
Root an. cond. $6.36271$
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  + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s + 12·23-s − 25-s + 4·27-s − 3·36-s + 8·43-s + 2·48-s + 10·49-s − 12·53-s − 20·61-s − 64-s + 24·69-s − 2·75-s + 16·79-s + 5·81-s − 12·92-s + 100-s + 24·101-s − 16·103-s + 24·107-s − 4·108-s − 24·113-s + 22·121-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s + 2.50·23-s − 1/5·25-s + 0.769·27-s − 1/2·36-s + 1.21·43-s + 0.288·48-s + 10/7·49-s − 1.64·53-s − 2.56·61-s − 1/8·64-s + 2.88·69-s − 0.230·75-s + 1.80·79-s + 5/9·81-s − 1.25·92-s + 1/10·100-s + 2.38·101-s − 1.57·103-s + 2.32·107-s − 0.384·108-s − 2.25·113-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25704900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1638.96\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.400166609\)
\(L(\frac12)\) \(\approx\) \(4.400166609\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + T^{2} \)
13 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.453759769084378217737790441724, −7.957909397047282314732791807547, −7.72396099817686158039038024753, −7.56203604562913967617033268408, −6.91327415804241022931897801510, −6.88481688968352137943313621485, −6.15564634257488720778145523548, −6.05688339399078210721993952036, −5.24331532351623691715885429779, −5.18371594968828039866394849513, −4.55224464269995044246134600256, −4.44398402993999719145757551735, −3.83875701704384749159726300700, −3.48082916968044289119079336935, −2.92301842864586702579521695990, −2.89190716999261922645646820403, −2.21395341874504585781092200842, −1.66860415195555780406686943718, −1.12664987648233769261960092703, −0.57548045762163227066885860546, 0.57548045762163227066885860546, 1.12664987648233769261960092703, 1.66860415195555780406686943718, 2.21395341874504585781092200842, 2.89190716999261922645646820403, 2.92301842864586702579521695990, 3.48082916968044289119079336935, 3.83875701704384749159726300700, 4.44398402993999719145757551735, 4.55224464269995044246134600256, 5.18371594968828039866394849513, 5.24331532351623691715885429779, 6.05688339399078210721993952036, 6.15564634257488720778145523548, 6.88481688968352137943313621485, 6.91327415804241022931897801510, 7.56203604562913967617033268408, 7.72396099817686158039038024753, 7.957909397047282314732791807547, 8.453759769084378217737790441724

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