Properties

Label 4-1568e2-1.1-c1e2-0-31
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $156.763$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 3·9-s − 10·17-s − 7·25-s + 16·29-s − 10·37-s − 8·41-s + 6·45-s − 2·53-s − 22·61-s − 30·73-s + 20·85-s − 14·89-s − 24·97-s + 22·101-s − 6·109-s + 8·113-s + 5·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s + 30·153-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s − 2.42·17-s − 7/5·25-s + 2.97·29-s − 1.64·37-s − 1.24·41-s + 0.894·45-s − 0.274·53-s − 2.81·61-s − 3.51·73-s + 2.16·85-s − 1.48·89-s − 2.43·97-s + 2.18·101-s − 0.574·109-s + 0.752·113-s + 5/11·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(156.763\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1568} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 155 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−8.889787970155161537597042452007, −8.832443402605427668192012600181, −8.417381061860846762295891042862, −8.293127084651543003213691605111, −7.62473822803488943409818211173, −7.21281786989420387676862329829, −6.85795088417944299202644685591, −6.37383405352651148768533478815, −6.06212653916999263362136602376, −5.62796939912276000558408700129, −4.81553158730289397814414536046, −4.67101370445417715055346440926, −4.24807379718754068151567908632, −3.71459672137746628721843940476, −2.95494982248601996920182025182, −2.87381173586581413683708153789, −2.04633501668482838765722432535, −1.44109671263742063634594123878, 0, 0, 1.44109671263742063634594123878, 2.04633501668482838765722432535, 2.87381173586581413683708153789, 2.95494982248601996920182025182, 3.71459672137746628721843940476, 4.24807379718754068151567908632, 4.67101370445417715055346440926, 4.81553158730289397814414536046, 5.62796939912276000558408700129, 6.06212653916999263362136602376, 6.37383405352651148768533478815, 6.85795088417944299202644685591, 7.21281786989420387676862329829, 7.62473822803488943409818211173, 8.293127084651543003213691605111, 8.417381061860846762295891042862, 8.832443402605427668192012600181, 8.889787970155161537597042452007

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