Properties

Label 4-1216e2-1.1-c3e2-0-8
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $5147.53$
Root an. cond. $8.47032$
Motivic weight $3$
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  + 5·3-s + 5·5-s − 30·7-s − 27·9-s + 71·11-s + 35·13-s + 25·15-s + 38·19-s − 150·21-s − 5·23-s − 25·25-s − 260·27-s − 155·29-s − 88·31-s + 355·33-s − 150·35-s − 380·37-s + 175·39-s − 142·41-s − 155·43-s − 135·45-s − 455·47-s + 121·49-s + 275·53-s + 355·55-s + 190·57-s + 873·59-s + ⋯
L(s)  = 1  + 0.962·3-s + 0.447·5-s − 1.61·7-s − 9-s + 1.94·11-s + 0.746·13-s + 0.430·15-s + 0.458·19-s − 1.55·21-s − 0.0453·23-s − 1/5·25-s − 1.85·27-s − 0.992·29-s − 0.509·31-s + 1.87·33-s − 0.724·35-s − 1.68·37-s + 0.718·39-s − 0.540·41-s − 0.549·43-s − 0.447·45-s − 1.41·47-s + 0.352·49-s + 0.712·53-s + 0.870·55-s + 0.441·57-s + 1.92·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5147.53\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
good3$D_{4}$ \( 1 - 5 T + 52 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - p T + 2 p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 30 T + 779 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 71 T + 3716 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 35 T + 3702 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9529 T^{2} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 5 T - 4378 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 155 T + 19928 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 88 T + 8718 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 380 T + 136878 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 142 T + 135458 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 155 T + 52086 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 455 T + 255038 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 275 T + 191846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 873 T + 591184 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 445 T + 250812 T^{2} + 445 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 645 T + 674834 T^{2} + 645 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1712 T + 1445258 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 1274 T + 1391022 T^{2} + 1274 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 90 T + 1038382 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 888 T + 1554274 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 710 T + 220818 T^{2} - 710 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.068167005174113828773854213085, −8.946069198867718019916370215626, −8.349946623726534446920066140609, −8.319150169414500960058484808155, −7.25646776305200764167941850661, −7.16503793920054272441411870222, −6.70631343420272650918033110902, −6.13257288915913259097730484378, −5.83162928632214186650586225155, −5.69028292856084874870147978156, −4.83115157832770512854110309056, −4.15944903477114194023122068929, −3.53287340085859216951421719995, −3.44520716762330185605204366758, −3.08138258551269292699289428468, −2.42595116677216165574092280471, −1.59811692379856497312730613289, −1.39475089522996620097633980418, 0, 0, 1.39475089522996620097633980418, 1.59811692379856497312730613289, 2.42595116677216165574092280471, 3.08138258551269292699289428468, 3.44520716762330185605204366758, 3.53287340085859216951421719995, 4.15944903477114194023122068929, 4.83115157832770512854110309056, 5.69028292856084874870147978156, 5.83162928632214186650586225155, 6.13257288915913259097730484378, 6.70631343420272650918033110902, 7.16503793920054272441411870222, 7.25646776305200764167941850661, 8.319150169414500960058484808155, 8.349946623726534446920066140609, 8.946069198867718019916370215626, 9.068167005174113828773854213085

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