Properties

Label 4-1216e2-1.1-c1e2-0-14
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·5-s + 4·7-s + 3·9-s − 6·11-s + 2·13-s − 6·15-s + 6·17-s + 7·19-s − 12·21-s − 4·23-s + 5·25-s − 6·29-s + 16·31-s + 18·33-s + 8·35-s + 16·37-s − 6·39-s + 3·41-s + 8·43-s + 6·45-s + 10·47-s − 2·49-s − 18·51-s + 2·53-s − 12·55-s − 21·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s + 1.51·7-s + 9-s − 1.80·11-s + 0.554·13-s − 1.54·15-s + 1.45·17-s + 1.60·19-s − 2.61·21-s − 0.834·23-s + 25-s − 1.11·29-s + 2.87·31-s + 3.13·33-s + 1.35·35-s + 2.63·37-s − 0.960·39-s + 0.468·41-s + 1.21·43-s + 0.894·45-s + 1.45·47-s − 2/7·49-s − 2.52·51-s + 0.274·53-s − 1.61·55-s − 2.78·57-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

\(L(1)\) \(\approx\) \(1.911417997\)
\(L(\frac12)\) \(\approx\) \(1.911417997\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 3 T - 88 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−9.958842003803732604735158127500, −9.831357223942675822520507549263, −9.305751994251309825856240194762, −8.434182535856960275161287202316, −8.370900778444348654450894934352, −7.77685665908670027529401752847, −7.44951945779846656210046650170, −7.25644334482054619575657468013, −6.18783085175458604362242869517, −5.94174340944807833801977567585, −5.77479697604734160832670129259, −5.46984632442411190606259456080, −4.85545938855094899517780733150, −4.73169985869724791905039860160, −4.11322932448793540072501119738, −3.10484149135912992263983793789, −2.69558462052687484145779530331, −2.06976718773429540194391168478, −1.02148054308135995621870827752, −0.886290636852528753201321402388, 0.886290636852528753201321402388, 1.02148054308135995621870827752, 2.06976718773429540194391168478, 2.69558462052687484145779530331, 3.10484149135912992263983793789, 4.11322932448793540072501119738, 4.73169985869724791905039860160, 4.85545938855094899517780733150, 5.46984632442411190606259456080, 5.77479697604734160832670129259, 5.94174340944807833801977567585, 6.18783085175458604362242869517, 7.25644334482054619575657468013, 7.44951945779846656210046650170, 7.77685665908670027529401752847, 8.370900778444348654450894934352, 8.434182535856960275161287202316, 9.305751994251309825856240194762, 9.831357223942675822520507549263, 9.958842003803732604735158127500

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