Properties

Label 4-3920e2-1.1-c1e2-0-8
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
Motivic weight $1$
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  − 2·3-s − 2·5-s − 9-s + 2·11-s + 2·13-s + 4·15-s + 2·17-s − 4·19-s − 4·23-s + 3·25-s + 6·27-s + 2·29-s − 12·31-s − 4·33-s + 4·37-s − 4·39-s + 12·41-s + 12·43-s + 2·45-s − 18·47-s − 4·51-s + 16·53-s − 4·55-s + 8·57-s − 12·59-s − 8·61-s − 4·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 1/3·9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 3/5·25-s + 1.15·27-s + 0.371·29-s − 2.15·31-s − 0.696·33-s + 0.657·37-s − 0.640·39-s + 1.87·41-s + 1.82·43-s + 0.298·45-s − 2.62·47-s − 0.560·51-s + 2.19·53-s − 0.539·55-s + 1.05·57-s − 1.56·59-s − 1.02·61-s − 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T - 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 18 T + 173 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 16 T + 168 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 132 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 207 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 14 T + 193 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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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

−8.212202085364256999871679890603, −7.81780668549322091192010616884, −7.44968148958444710347092242673, −7.43643064377554768369987889943, −6.61293572037851786980048403129, −6.32334026424503463920268573240, −6.00821793814334059263047060638, −5.92583084403926200635485365510, −5.18476604129672236553736385818, −5.10287439619040863164428582986, −4.36038736261988811124027475090, −4.16222448319663390693613891136, −3.69859335946998079718905254593, −3.41016256350051918646629567689, −2.62213510511166818472869257798, −2.38716136296809677141122983247, −1.42259025245414493332491213180, −1.09650746697329169570210073272, 0, 0, 1.09650746697329169570210073272, 1.42259025245414493332491213180, 2.38716136296809677141122983247, 2.62213510511166818472869257798, 3.41016256350051918646629567689, 3.69859335946998079718905254593, 4.16222448319663390693613891136, 4.36038736261988811124027475090, 5.10287439619040863164428582986, 5.18476604129672236553736385818, 5.92583084403926200635485365510, 6.00821793814334059263047060638, 6.32334026424503463920268573240, 6.61293572037851786980048403129, 7.43643064377554768369987889943, 7.44968148958444710347092242673, 7.81780668549322091192010616884, 8.212202085364256999871679890603

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