Properties

Label 4-1216e2-1.1-c3e2-0-5
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

−9.149395680478909961097795249810, −8.509709102810924684702088452231, −8.276986949101237224020711997632, −8.158163011625687449257356833488, −7.50390804945420585177821731749, −7.21877894849149817391165167785, −6.34738715232421568336544774518, −6.26020257736497502214071741744, −5.50771345691728118716779625739, −5.40258529629632205782125147738, −5.08966353095973525499499955594, −4.71329461459773408940567503127, −3.95619542069182722002231512217, −3.43040795104992083915289705758, −2.53370931477470693213653530024, −2.45645241780863011546914379634, −1.63784662278662505139340918555, −1.08335597844511349240491141856, 0, 0, 1.08335597844511349240491141856, 1.63784662278662505139340918555, 2.45645241780863011546914379634, 2.53370931477470693213653530024, 3.43040795104992083915289705758, 3.95619542069182722002231512217, 4.71329461459773408940567503127, 5.08966353095973525499499955594, 5.40258529629632205782125147738, 5.50771345691728118716779625739, 6.26020257736497502214071741744, 6.34738715232421568336544774518, 7.21877894849149817391165167785, 7.50390804945420585177821731749, 8.158163011625687449257356833488, 8.276986949101237224020711997632, 8.509709102810924684702088452231, 9.149395680478909961097795249810

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