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Properties

Label 4-538e2-1.1-c1e2-0-0
Degree $4$
Conductor $289444$
Sign $1$
Analytic cond. $18.4552$
Root an. cond. $2.07266$
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  + 2·2-s + 3-s + 3·4-s + 5-s + 2·6-s − 2·7-s + 4·8-s − 2·9-s + 2·10-s + 6·11-s + 3·12-s + 10·13-s − 4·14-s + 15-s + 5·16-s − 4·17-s − 4·18-s + 4·19-s + 3·20-s − 2·21-s + 12·22-s + 2·23-s + 4·24-s − 6·25-s + 20·26-s − 2·27-s − 6·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s + 1.41·8-s − 2/3·9-s + 0.632·10-s + 1.80·11-s + 0.866·12-s + 2.77·13-s − 1.06·14-s + 0.258·15-s + 5/4·16-s − 0.970·17-s − 0.942·18-s + 0.917·19-s + 0.670·20-s − 0.436·21-s + 2.55·22-s + 0.417·23-s + 0.816·24-s − 6/5·25-s + 3.92·26-s − 0.384·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(289444\)    =    \(2^{2} \cdot 269^{2}\)
Sign: $1$
Analytic conductor: \(18.4552\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 289444,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.300699560\)
\(L(\frac12)\) \(\approx\) \(6.300699560\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
269$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 13 T + 97 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 9 T + 85 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 18 T + 163 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \)
79$C_4$ \( 1 + 9 T + 175 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 11 T + 205 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 182 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−11.14746901950943647195748594240, −10.95106837881552665093781446547, −10.12074581127786568131326877850, −9.776204173864185958574316016086, −9.215065162740532783467983571224, −8.801734598099893482141580861158, −8.370356655563098782338215071683, −8.129152520931152539430414403722, −7.08655305238369845374278922641, −6.59432356543163022882436908892, −6.45769168146058318023883440281, −6.11355403001466061285262485649, −5.51402124410881792477446818018, −4.94760952852127934613324027835, −4.20321158622666857924280124019, −3.62739899889701253609734948047, −3.37919496802167730530231627105, −3.01077873099390255308634150019, −1.79846874068588125902600475991, −1.42553369830711364388203249635, 1.42553369830711364388203249635, 1.79846874068588125902600475991, 3.01077873099390255308634150019, 3.37919496802167730530231627105, 3.62739899889701253609734948047, 4.20321158622666857924280124019, 4.94760952852127934613324027835, 5.51402124410881792477446818018, 6.11355403001466061285262485649, 6.45769168146058318023883440281, 6.59432356543163022882436908892, 7.08655305238369845374278922641, 8.129152520931152539430414403722, 8.370356655563098782338215071683, 8.801734598099893482141580861158, 9.215065162740532783467983571224, 9.776204173864185958574316016086, 10.12074581127786568131326877850, 10.95106837881552665093781446547, 11.14746901950943647195748594240

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