Properties

Label 2-7623-1.1-c1-0-75
Degree $2$
Conductor $7623$
Sign $1$
Analytic cond. $60.8699$
Root an. cond. $7.80192$
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-s − 4-s − 5-s + 7-s − 3·8-s − 10-s + 5·13-s + 14-s − 16-s + 7·17-s − 6·19-s + 20-s + 4·23-s − 4·25-s + 5·26-s − 28-s − 9·29-s − 2·31-s + 5·32-s + 7·34-s − 35-s + 9·37-s − 6·38-s + 3·40-s + 7·41-s − 6·43-s + 4·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.316·10-s + 1.38·13-s + 0.267·14-s − 1/4·16-s + 1.69·17-s − 1.37·19-s + 0.223·20-s + 0.834·23-s − 4/5·25-s + 0.980·26-s − 0.188·28-s − 1.67·29-s − 0.359·31-s + 0.883·32-s + 1.20·34-s − 0.169·35-s + 1.47·37-s − 0.973·38-s + 0.474·40-s + 1.09·41-s − 0.914·43-s + 0.589·46-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: $1$
Analytic conductor: \(60.8699\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.195090343\)
\(L(\frac12)\) \(\approx\) \(2.195090343\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−7.924542437429132948347184150017, −7.25461409171001655086026789468, −6.00951301896000859750911899657, −5.92767608270396503684304442532, −4.99515890785368532494232706062, −4.19904675844402256642721198697, −3.70766736164884112751416686431, −3.05948199405770213339884044020, −1.78639229445410984565034470782, −0.68303465491592391562695088684, 0.68303465491592391562695088684, 1.78639229445410984565034470782, 3.05948199405770213339884044020, 3.70766736164884112751416686431, 4.19904675844402256642721198697, 4.99515890785368532494232706062, 5.92767608270396503684304442532, 6.00951301896000859750911899657, 7.25461409171001655086026789468, 7.924542437429132948347184150017

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