Properties

Label 2-361920-1.1-c1-0-40
Degree $2$
Conductor $361920$
Sign $1$
Analytic cond. $2889.94$
Root an. cond. $53.7582$
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  + 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 13-s − 15-s − 6·17-s + 4·19-s − 4·21-s − 8·23-s + 25-s + 27-s − 29-s + 8·31-s + 4·33-s + 4·35-s + 6·37-s − 39-s − 2·41-s + 4·43-s − 45-s + 9·49-s − 6·51-s + 10·53-s − 4·55-s + 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 0.986·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.840·51-s + 1.37·53-s − 0.539·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361920\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 29\)
Sign: $1$
Analytic conductor: \(2889.94\)
Root analytic conductor: \(53.7582\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{361920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 361920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.841962918\)
\(L(\frac12)\) \(\approx\) \(2.841962918\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
29 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.66978218325471, −11.95922751048360, −11.61011525221364, −11.48970142883362, −10.49264379449074, −10.17084646857274, −9.727885643091783, −9.361552809330974, −8.932267132903384, −8.514101052395134, −7.925081405155723, −7.491378051958935, −6.820807230013759, −6.606877297888649, −6.215088996130506, −5.620613263530568, −4.891153304142779, −4.203066774872060, −3.944441625314745, −3.556383533689028, −2.886228814974733, −2.378984181699069, −1.930756897377769, −0.8605783229031451, −0.5317289855231084, 0.5317289855231084, 0.8605783229031451, 1.930756897377769, 2.378984181699069, 2.886228814974733, 3.556383533689028, 3.944441625314745, 4.203066774872060, 4.891153304142779, 5.620613263530568, 6.215088996130506, 6.606877297888649, 6.820807230013759, 7.491378051958935, 7.925081405155723, 8.514101052395134, 8.932267132903384, 9.361552809330974, 9.727885643091783, 10.17084646857274, 10.49264379449074, 11.48970142883362, 11.61011525221364, 11.95922751048360, 12.66978218325471

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