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Properties

Label 2-2523-87.62-c0-0-5
Degree $2$
Conductor $2523$
Sign $-0.620 + 0.784i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 + 0.433i)3-s + (0.222 + 0.974i)6-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + (−0.623 − 0.781i)13-s + (−0.222 − 0.974i)14-s + (−0.623 − 0.781i)16-s − 17-s + (−0.623 − 0.781i)18-s + (−0.623 + 0.781i)21-s + (−0.900 + 0.433i)22-s + (−0.222 + 0.974i)24-s + (−0.900 − 0.433i)25-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 + 0.433i)3-s + (0.222 + 0.974i)6-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + (−0.623 − 0.781i)13-s + (−0.222 − 0.974i)14-s + (−0.623 − 0.781i)16-s − 17-s + (−0.623 − 0.781i)18-s + (−0.623 + 0.781i)21-s + (−0.900 + 0.433i)22-s + (−0.222 + 0.974i)24-s + (−0.900 − 0.433i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.620 + 0.784i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -0.620 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060812333\)
\(L(\frac12)\) \(\approx\) \(1.060812333\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + (-0.623 - 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.900 + 0.433i)T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (0.222 + 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
97 \( 1 + (-0.623 - 0.781i)T^{2} \)
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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

−9.033878809438683213377161669992, −7.932911498695031315303581693445, −7.37706393348803559047307699322, −6.36940297298677776158953201502, −5.51809289880232653363095168826, −4.66052432160282971639365041154, −4.08251068649959922574677290303, −3.04929513133707638050888393688, −2.02259850046972205866861271184, −0.70579619196620752863530965021, 1.78579715430193455355742080350, 2.30439759405006232302756098277, 4.48435887671298128320063988908, 4.74315348688673940598754381566, 5.62286600697154549283798732531, 6.24315205151083145641893286134, 7.10827233571128393416310805826, 7.55163465614431997442510363719, 8.207193014963854344308496058020, 9.259611215382751912348011141194

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