Properties

Label 4-1755e2-1.1-c1e2-0-10
Degree $4$
Conductor $3080025$
Sign $1$
Analytic cond. $196.385$
Root an. cond. $3.74349$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·4-s − 5-s − 2·7-s − 5·8-s + 10-s − 11-s + 13-s + 2·14-s + 5·16-s − 4·17-s − 6·19-s − 2·20-s + 22-s − 26-s − 4·28-s + 5·29-s + 31-s − 10·32-s + 4·34-s + 2·35-s − 10·37-s + 6·38-s + 5·40-s + 8·43-s − 2·44-s − 2·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 0.447·5-s − 0.755·7-s − 1.76·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 0.534·14-s + 5/4·16-s − 0.970·17-s − 1.37·19-s − 0.447·20-s + 0.213·22-s − 0.196·26-s − 0.755·28-s + 0.928·29-s + 0.179·31-s − 1.76·32-s + 0.685·34-s + 0.338·35-s − 1.64·37-s + 0.973·38-s + 0.790·40-s + 1.21·43-s − 0.301·44-s − 0.291·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3080025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3080025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3080025\)    =    \(3^{6} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(196.385\)
Root analytic conductor: \(3.74349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3080025,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 10 T + 17 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−9.008135041586029053512543974070, −8.894099906779463724199322950964, −8.240372061130828114126407554162, −8.065496853768385230930604172136, −7.63916520216298262106241334035, −7.00117035277627246441829788683, −6.62093839365958432014770772621, −6.46163334418971755662590960971, −6.17705086673578182753206858978, −5.58341290204056281625952393212, −5.09667498073455144308057212363, −4.42325939299105524856234472394, −4.07926188039614167590725058724, −3.37463358066403399753786183922, −3.03091185132671330206734504038, −2.57977611524376115471689126586, −2.03028915539389220282682487604, −1.32547714595512309609106092183, 0, 0, 1.32547714595512309609106092183, 2.03028915539389220282682487604, 2.57977611524376115471689126586, 3.03091185132671330206734504038, 3.37463358066403399753786183922, 4.07926188039614167590725058724, 4.42325939299105524856234472394, 5.09667498073455144308057212363, 5.58341290204056281625952393212, 6.17705086673578182753206858978, 6.46163334418971755662590960971, 6.62093839365958432014770772621, 7.00117035277627246441829788683, 7.63916520216298262106241334035, 8.065496853768385230930604172136, 8.240372061130828114126407554162, 8.894099906779463724199322950964, 9.008135041586029053512543974070

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