Properties

Degree $2$
Conductor $7623$
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 − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 2·13-s − 14-s − 16-s − 6·17-s − 4·19-s − 2·20-s − 25-s − 2·26-s − 28-s − 2·29-s − 5·32-s + 6·34-s + 2·35-s + 6·37-s + 4·38-s + 6·40-s + 2·41-s + 4·43-s + 49-s + 50-s − 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.883·32-s + 1.02·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.141·50-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7623} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.256465797\)
\(L(\frac12)\) \(\approx\) \(1.256465797\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−7.976525779912433683772333851675, −7.39398850413165927730875614743, −6.34518093653275179095390844834, −6.01581061696917710851209529028, −4.92403553365775964167265796866, −4.46792640657225865455095535714, −3.62088779784825026995011416777, −2.28059162307602342837183603744, −1.77611296934943774735876447029, −0.63334663369662260021570161928, 0.63334663369662260021570161928, 1.77611296934943774735876447029, 2.28059162307602342837183603744, 3.62088779784825026995011416777, 4.46792640657225865455095535714, 4.92403553365775964167265796866, 6.01581061696917710851209529028, 6.34518093653275179095390844834, 7.39398850413165927730875614743, 7.976525779912433683772333851675

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