Properties

Label 4-3920e2-1.1-c1e2-0-6
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 $0$

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: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.880399544\)
\(L(\frac12)\) \(\approx\) \(5.880399544\)
\(L(\frac{3}{2})\) 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

−8.700256457131114110362495517554, −8.371188211186711246665722215860, −8.008238389993279901289224483510, −7.67093056227453065762255424778, −7.12797556954179945664523728680, −6.95058862883532081958357124370, −6.31437896465821692875260932167, −6.23470670692480588538784654503, −5.57598826194398746727814890718, −5.39536539034783086301336818665, −4.97038103860650556036123884087, −4.35979424550451549107269334042, −3.87322168122813769653887350392, −3.78756830819364628679704000857, −2.90923026907455797576820958185, −2.69344984782767908623885839015, −2.33441800127376516481979362904, −2.04870677627656162068820210702, −1.08188887374013321316667784630, −0.73080096487966109812042463586, 0.73080096487966109812042463586, 1.08188887374013321316667784630, 2.04870677627656162068820210702, 2.33441800127376516481979362904, 2.69344984782767908623885839015, 2.90923026907455797576820958185, 3.78756830819364628679704000857, 3.87322168122813769653887350392, 4.35979424550451549107269334042, 4.97038103860650556036123884087, 5.39536539034783086301336818665, 5.57598826194398746727814890718, 6.23470670692480588538784654503, 6.31437896465821692875260932167, 6.95058862883532081958357124370, 7.12797556954179945664523728680, 7.67093056227453065762255424778, 8.008238389993279901289224483510, 8.371188211186711246665722215860, 8.700256457131114110362495517554

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