Properties

Label 4-800320-1.1-c1e2-0-2
Degree $4$
Conductor $800320$
Sign $1$
Analytic cond. $51.0290$
Root an. cond. $2.67272$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 2·4-s + 3·5-s + 4·6-s + 2·7-s − 9-s + 6·10-s − 4·11-s + 4·12-s + 4·13-s + 4·14-s + 6·15-s − 4·16-s + 3·17-s − 2·18-s + 6·20-s + 4·21-s − 8·22-s + 8·25-s + 8·26-s − 6·27-s + 4·28-s + 4·29-s + 12·30-s + 2·31-s − 8·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s + 1.34·5-s + 1.63·6-s + 0.755·7-s − 1/3·9-s + 1.89·10-s − 1.20·11-s + 1.15·12-s + 1.10·13-s + 1.06·14-s + 1.54·15-s − 16-s + 0.727·17-s − 0.471·18-s + 1.34·20-s + 0.872·21-s − 1.70·22-s + 8/5·25-s + 1.56·26-s − 1.15·27-s + 0.755·28-s + 0.742·29-s + 2.19·30-s + 0.359·31-s − 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(800320\)    =    \(2^{6} \cdot 5 \cdot 41 \cdot 61\)
Sign: $1$
Analytic conductor: \(51.0290\)
Root analytic conductor: \(2.67272\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{800320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 800320,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.724615788\)
\(L(\frac12)\) \(\approx\) \(8.724615788\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 12 T + p T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 7 T + p T^{2} ) \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 13 T + 88 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T - 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 42 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 132 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T - 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 139 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T - 57 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−12.5270659603, −12.0005097693, −11.5054251596, −11.2202683780, −10.8380279619, −10.2147704861, −10.0004781736, −9.44733053991, −9.05090927347, −8.48407259275, −8.24525376641, −8.06214566417, −7.31750930575, −6.66269687918, −6.33455748286, −5.82869171309, −5.45816837729, −5.01599041629, −4.74711116255, −3.88450241261, −3.42269921427, −2.90069426596, −2.49015115619, −2.07870064030, −1.13269701141, 1.13269701141, 2.07870064030, 2.49015115619, 2.90069426596, 3.42269921427, 3.88450241261, 4.74711116255, 5.01599041629, 5.45816837729, 5.82869171309, 6.33455748286, 6.66269687918, 7.31750930575, 8.06214566417, 8.24525376641, 8.48407259275, 9.05090927347, 9.44733053991, 10.0004781736, 10.2147704861, 10.8380279619, 11.2202683780, 11.5054251596, 12.0005097693, 12.5270659603

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