Properties

Label 2-76664-1.1-c1-0-4
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  + 2·3-s + 2·5-s − 7-s + 9-s − 5·11-s + 6·13-s + 4·15-s − 2·17-s − 4·19-s − 2·21-s − 3·23-s − 25-s − 4·27-s + 7·29-s + 10·31-s − 10·33-s − 2·35-s + 12·39-s + 10·41-s + 4·43-s + 2·45-s + 6·47-s + 49-s − 4·51-s + 3·53-s − 10·55-s − 8·57-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.66·13-s + 1.03·15-s − 0.485·17-s − 0.917·19-s − 0.436·21-s − 0.625·23-s − 1/5·25-s − 0.769·27-s + 1.29·29-s + 1.79·31-s − 1.74·33-s − 0.338·35-s + 1.92·39-s + 1.56·41-s + 0.609·43-s + 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.560·51-s + 0.412·53-s − 1.34·55-s − 1.05·57-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

\(L(1)\) \(\approx\) \(3.909981796\)
\(L(\frac12)\) \(\approx\) \(3.909981796\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.93463013710154, −13.68522876495772, −13.16221588459334, −12.84612547363203, −12.22971988784853, −11.47452731047165, −10.83540712296314, −10.37158699795994, −10.15921644697497, −9.204476851865157, −9.142039410541864, −8.380888594512539, −8.045049816494892, −7.693247855249279, −6.687267669556594, −6.204978016075898, −5.904931686868564, −5.223229219522413, −4.298747585386728, −4.035195667477618, −3.056382716192067, −2.676896963440905, −2.275086759724910, −1.500469650241644, −0.5798397516615105, 0.5798397516615105, 1.500469650241644, 2.275086759724910, 2.676896963440905, 3.056382716192067, 4.035195667477618, 4.298747585386728, 5.223229219522413, 5.904931686868564, 6.204978016075898, 6.687267669556594, 7.693247855249279, 8.045049816494892, 8.380888594512539, 9.142039410541864, 9.204476851865157, 10.15921644697497, 10.37158699795994, 10.83540712296314, 11.47452731047165, 12.22971988784853, 12.84612547363203, 13.16221588459334, 13.68522876495772, 13.93463013710154

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