Properties

Label 4-1216e2-1.1-c1e2-0-5
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
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  + 2·5-s − 6·9-s + 14·17-s − 7·25-s − 12·45-s − 5·49-s − 30·61-s + 22·73-s + 27·81-s + 28·85-s + 20·101-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 84·153-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s + 3.39·17-s − 7/5·25-s − 1.78·45-s − 5/7·49-s − 3.84·61-s + 2.57·73-s + 3·81-s + 3.03·85-s + 1.99·101-s + 3/11·121-s − 2.32·125-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 − 6.79·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.014351525\)
\(L(\frac12)\) \(\approx\) \(2.014351525\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−9.733159759828253484517382507585, −9.598640470651873021927810079484, −9.289689289756409509468176085632, −8.730652112905267376001911698192, −8.182529881711780159313974762446, −7.892461696995965956192970964650, −7.72824453972356422499019613261, −7.14724539298870820808912222030, −6.27600795084745381564055865545, −6.05933519784711203569568899208, −5.84570053584007553397729896467, −5.27621839607647452944859918194, −5.19137696551245104377824290314, −4.36506464712612395828591044825, −3.44894845950579846685311982810, −3.33993944117661109114552992065, −2.88082250231297329570457088281, −2.10492350651637610444647579425, −1.54550165955808104194140134970, −0.61090724662325347051808232074, 0.61090724662325347051808232074, 1.54550165955808104194140134970, 2.10492350651637610444647579425, 2.88082250231297329570457088281, 3.33993944117661109114552992065, 3.44894845950579846685311982810, 4.36506464712612395828591044825, 5.19137696551245104377824290314, 5.27621839607647452944859918194, 5.84570053584007553397729896467, 6.05933519784711203569568899208, 6.27600795084745381564055865545, 7.14724539298870820808912222030, 7.72824453972356422499019613261, 7.892461696995965956192970964650, 8.182529881711780159313974762446, 8.730652112905267376001911698192, 9.289689289756409509468176085632, 9.598640470651873021927810079484, 9.733159759828253484517382507585

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