Properties

Degree $2$
Conductor $40362$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 2·5-s + 6-s − 7-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s − 6·13-s − 14-s − 2·15-s + 16-s − 2·17-s + 18-s − 4·19-s − 2·20-s − 21-s + 4·22-s − 8·23-s + 24-s − 25-s − 6·26-s + 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40362 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40362 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40362\)    =    \(2 \cdot 3 \cdot 7 \cdot 31^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{40362} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 40362,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.265881482\)
\(L(\frac12)\) \(\approx\) \(2.265881482\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 + T \)
31 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.72999444821970, −14.28600728934306, −13.91870272154273, −13.19422523251217, −12.58816344414239, −12.30476976116348, −11.66287112679683, −11.48414794851402, −10.54131915331042, −10.03230850166984, −9.485248350350337, −8.993976957211578, −8.140050040613731, −7.838541721326751, −7.239982823129531, −6.544290280316228, −6.284633260559943, −5.372799230047240, −4.478639224455680, −4.239372965887268, −3.758268746119678, −2.952217862100299, −2.326257412340739, −1.714800699927614, −0.4437064223227429, 0.4437064223227429, 1.714800699927614, 2.326257412340739, 2.952217862100299, 3.758268746119678, 4.239372965887268, 4.478639224455680, 5.372799230047240, 6.284633260559943, 6.544290280316228, 7.239982823129531, 7.838541721326751, 8.140050040613731, 8.993976957211578, 9.485248350350337, 10.03230850166984, 10.54131915331042, 11.48414794851402, 11.66287112679683, 12.30476976116348, 12.58816344414239, 13.19422523251217, 13.91870272154273, 14.28600728934306, 14.72999444821970

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