Properties

Label 4-3920e2-1.1-c1e2-0-9
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  − 4·3-s + 2·5-s + 8·9-s − 4·11-s − 4·13-s − 8·15-s + 8·17-s − 4·19-s + 8·23-s + 3·25-s − 12·27-s − 4·29-s + 16·33-s − 4·37-s + 16·39-s + 8·41-s + 12·43-s + 16·45-s − 16·47-s − 32·51-s − 4·53-s − 8·55-s + 16·57-s − 20·59-s − 4·61-s − 8·65-s + 8·67-s + ⋯
L(s)  = 1  − 2.30·3-s + 0.894·5-s + 8/3·9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s + 1.94·17-s − 0.917·19-s + 1.66·23-s + 3/5·25-s − 2.30·27-s − 0.742·29-s + 2.78·33-s − 0.657·37-s + 2.56·39-s + 1.24·41-s + 1.82·43-s + 2.38·45-s − 2.33·47-s − 4.48·51-s − 0.549·53-s − 1.07·55-s + 2.11·57-s − 2.60·59-s − 0.512·61-s − 0.992·65-s + 0.977·67-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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 152 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
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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

−8.004244852742722880106199917132, −7.75725520800288412737175554622, −7.53523454045351451531132927466, −7.15251506997773743161281959086, −6.53516021015352282878328312876, −6.43660811306357188989470610103, −5.79959246322817348060878706750, −5.78767811224341639216595916388, −5.24980156820310574439749172021, −5.19131395441803621721540178762, −4.66307881814671215048982783548, −4.49212750388630077394297344160, −3.67165968245932620244619209745, −3.09705660914852860856788341149, −2.71518107405165525246788488776, −2.19287258485472466293923471795, −1.35597280925506657523864388447, −1.15834579324384561015485594743, 0, 0, 1.15834579324384561015485594743, 1.35597280925506657523864388447, 2.19287258485472466293923471795, 2.71518107405165525246788488776, 3.09705660914852860856788341149, 3.67165968245932620244619209745, 4.49212750388630077394297344160, 4.66307881814671215048982783548, 5.19131395441803621721540178762, 5.24980156820310574439749172021, 5.78767811224341639216595916388, 5.79959246322817348060878706750, 6.43660811306357188989470610103, 6.53516021015352282878328312876, 7.15251506997773743161281959086, 7.53523454045351451531132927466, 7.75725520800288412737175554622, 8.004244852742722880106199917132

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