Properties

Label 4-2925e2-1.1-c1e2-0-6
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $545.514$
Root an. cond. $4.83282$
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  + 3·4-s + 8·11-s + 5·16-s − 12·19-s − 8·29-s − 20·31-s + 12·41-s + 24·44-s + 10·49-s + 24·59-s + 4·61-s + 3·64-s − 36·76-s − 24·79-s + 28·89-s − 24·101-s + 4·109-s − 24·116-s + 26·121-s − 60·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 3/2·4-s + 2.41·11-s + 5/4·16-s − 2.75·19-s − 1.48·29-s − 3.59·31-s + 1.87·41-s + 3.61·44-s + 10/7·49-s + 3.12·59-s + 0.512·61-s + 3/8·64-s − 4.12·76-s − 2.70·79-s + 2.96·89-s − 2.38·101-s + 0.383·109-s − 2.22·116-s + 2.36·121-s − 5.38·124-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 + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(545.514\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.718986405\)
\(L(\frac12)\) \(\approx\) \(3.718986405\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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

−8.904400100217382714915692269242, −8.640478354764349830338426811408, −8.330534744689757208337144450049, −7.47341377614696586697224382353, −7.38190372289339276174273781044, −7.07784595736577217867491647207, −6.71751492774084447385584828445, −6.28199089012103472046539940971, −6.07874853784744423193277160814, −5.61295205423045792346346692628, −5.34320139178100409240757325305, −4.43370217421671038091236394715, −3.99125849550683717147330295451, −3.81546704977612050231870385336, −3.61091765668983068671578109101, −2.55928115053030425957022667095, −2.33229378177938181478096311341, −1.67968237057189886127633171735, −1.64514646517185367173134214557, −0.56667796819195692297450569715, 0.56667796819195692297450569715, 1.64514646517185367173134214557, 1.67968237057189886127633171735, 2.33229378177938181478096311341, 2.55928115053030425957022667095, 3.61091765668983068671578109101, 3.81546704977612050231870385336, 3.99125849550683717147330295451, 4.43370217421671038091236394715, 5.34320139178100409240757325305, 5.61295205423045792346346692628, 6.07874853784744423193277160814, 6.28199089012103472046539940971, 6.71751492774084447385584828445, 7.07784595736577217867491647207, 7.38190372289339276174273781044, 7.47341377614696586697224382353, 8.330534744689757208337144450049, 8.640478354764349830338426811408, 8.904400100217382714915692269242

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