Properties

Label 2-76664-1.1-c1-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s − 3·9-s − 4·11-s − 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s + 2·35-s + 2·41-s + 4·43-s + 6·45-s − 8·47-s + 49-s + 6·53-s + 8·55-s + 6·61-s + 3·63-s + 4·65-s − 4·67-s − 8·71-s + 10·73-s + 4·77-s − 16·79-s + 9·81-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 0.768·61-s + 0.377·63-s + 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + 81-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: \(1\)
Selberg data: \((2,\ 76664,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
7 \( 1 + T \)
37 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.57357680019283, −13.81579113709889, −13.06166721247249, −12.81506206553908, −12.40734098427201, −11.69171032489852, −11.40502030867923, −10.76445154295754, −10.36174947833556, −9.868752424856232, −9.092097018965172, −8.746876636603461, −7.962908757667455, −7.803525598059375, −7.294259936972061, −6.567350416921834, −5.816067127077831, −5.538557584580575, −4.922907012007806, −4.152186136691374, −3.617740588660897, −3.098785642057794, −2.414618237325888, −1.850443309628754, −0.5261698465860634, 0, 0.5261698465860634, 1.850443309628754, 2.414618237325888, 3.098785642057794, 3.617740588660897, 4.152186136691374, 4.922907012007806, 5.538557584580575, 5.816067127077831, 6.567350416921834, 7.294259936972061, 7.803525598059375, 7.962908757667455, 8.746876636603461, 9.092097018965172, 9.868752424856232, 10.36174947833556, 10.76445154295754, 11.40502030867923, 11.69171032489852, 12.40734098427201, 12.81506206553908, 13.06166721247249, 13.81579113709889, 14.57357680019283

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