Properties

Label 2-7623-1.1-c1-0-203
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 3·5-s − 7-s − 6·10-s + 2·13-s + 2·14-s − 4·16-s − 3·17-s − 4·19-s + 6·20-s − 2·23-s + 4·25-s − 4·26-s − 2·28-s − 8·29-s + 2·31-s + 8·32-s + 6·34-s − 3·35-s + 10·37-s + 8·38-s − 2·41-s + 9·43-s + 4·46-s − 9·47-s + 49-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.34·5-s − 0.377·7-s − 1.89·10-s + 0.554·13-s + 0.534·14-s − 16-s − 0.727·17-s − 0.917·19-s + 1.34·20-s − 0.417·23-s + 4/5·25-s − 0.784·26-s − 0.377·28-s − 1.48·29-s + 0.359·31-s + 1.41·32-s + 1.02·34-s − 0.507·35-s + 1.64·37-s + 1.29·38-s − 0.312·41-s + 1.37·43-s + 0.589·46-s − 1.31·47-s + 1/7·49-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: \(1\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.63590327162668529482196407477, −6.94942746030767269213041823496, −6.14935463529028811581800232597, −5.87901729810511972904163345294, −4.70997636001471276033834884131, −3.91961012314561270307060914469, −2.60879152012108403981305861661, −2.03323666012786418834526537363, −1.22575838935660182044456028025, 0, 1.22575838935660182044456028025, 2.03323666012786418834526537363, 2.60879152012108403981305861661, 3.91961012314561270307060914469, 4.70997636001471276033834884131, 5.87901729810511972904163345294, 6.14935463529028811581800232597, 6.94942746030767269213041823496, 7.63590327162668529482196407477

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