Properties

Label 2-76664-1.1-c1-0-1
Degree $2$
Conductor $76664$
Sign $1$
Analytic cond. $612.165$
Root an. cond. $24.7419$
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  − 3·5-s − 7-s − 3·9-s − 3·11-s + 5·13-s − 6·17-s − 4·19-s + 8·23-s + 4·25-s + 8·29-s + 7·31-s + 3·35-s + 2·41-s + 6·43-s + 9·45-s + 6·47-s + 49-s + 53-s + 9·55-s + 9·59-s − 6·61-s + 3·63-s − 15·65-s − 11·67-s − 15·71-s + 2·73-s + 3·77-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s − 9-s − 0.904·11-s + 1.38·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 4/5·25-s + 1.48·29-s + 1.25·31-s + 0.507·35-s + 0.312·41-s + 0.914·43-s + 1.34·45-s + 0.875·47-s + 1/7·49-s + 0.137·53-s + 1.21·55-s + 1.17·59-s − 0.768·61-s + 0.377·63-s − 1.86·65-s − 1.34·67-s − 1.78·71-s + 0.234·73-s + 0.341·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76664\)    =    \(2^{3} \cdot 7 \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(612.165\)
Root analytic conductor: \(24.7419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{76664} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 76664,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.131668988\)
\(L(\frac12)\) \(\approx\) \(1.131668988\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
37 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.85523850731623, −13.51786067885794, −13.11442692673266, −12.53067846543748, −11.99128390140725, −11.47600630572414, −11.07833728521297, −10.57661668203140, −10.40345593395838, −9.199337581594027, −8.777476574156996, −8.585804220121490, −7.978015298950400, −7.500953144101341, −6.697609061917847, −6.418043270151135, −5.799485635318221, −5.040199938961858, −4.399943743050849, −4.107684922376957, −3.190399324865566, −2.878651243991271, −2.256493535753619, −0.9893656007992680, −0.4234428036073524, 0.4234428036073524, 0.9893656007992680, 2.256493535753619, 2.878651243991271, 3.190399324865566, 4.107684922376957, 4.399943743050849, 5.040199938961858, 5.799485635318221, 6.418043270151135, 6.697609061917847, 7.500953144101341, 7.978015298950400, 8.585804220121490, 8.777476574156996, 9.199337581594027, 10.40345593395838, 10.57661668203140, 11.07833728521297, 11.47600630572414, 11.99128390140725, 12.53067846543748, 13.11442692673266, 13.51786067885794, 13.85523850731623

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