Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 29 \cdot 139 $
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 − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s − 2·11-s − 3·12-s − 7·13-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s − 19-s + 20-s − 3·21-s − 2·22-s − 4·23-s − 3·24-s + 25-s − 7·26-s − 9·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s − 0.866·12-s − 1.94·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.37·26-s − 1.73·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(40310\)    =    \(2 \cdot 5 \cdot 29 \cdot 139\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{40310} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 40310,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6269264199\)
\(L(\frac12)\)  \(\approx\)  \(0.6269264199\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;29,\;139\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;29,\;139\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
29 \( 1 - T \)
139 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.84040877228253, −14.20898241484622, −13.67734230965608, −13.03249272148460, −12.60724874665118, −12.15143843774710, −11.73115347145848, −11.28843573250233, −10.55317334807268, −10.33489868879431, −9.767235370715473, −9.172561336511538, −7.974244952020230, −7.781806894488222, −6.793758275259766, −6.614091592814273, −6.032368592091764, −5.197791545851071, −4.993428439952617, −4.665097664153090, −3.847837084831771, −2.792405009766381, −2.139205702698853, −1.483686724818508, −0.2750246464237492, 0.2750246464237492, 1.483686724818508, 2.139205702698853, 2.792405009766381, 3.847837084831771, 4.665097664153090, 4.993428439952617, 5.197791545851071, 6.032368592091764, 6.614091592814273, 6.793758275259766, 7.781806894488222, 7.974244952020230, 9.172561336511538, 9.767235370715473, 10.33489868879431, 10.55317334807268, 11.28843573250233, 11.73115347145848, 12.15143843774710, 12.60724874665118, 13.03249272148460, 13.67734230965608, 14.20898241484622, 14.84040877228253

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