Properties

Label 2-76664-1.1-c1-0-9
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  + 2·3-s + 5-s − 7-s + 9-s + 2·11-s − 2·13-s + 2·15-s + 17-s + 2·19-s − 2·21-s + 2·23-s − 4·25-s − 4·27-s − 3·29-s + 6·31-s + 4·33-s − 35-s − 4·39-s − 9·41-s − 8·43-s + 45-s + 6·47-s + 49-s + 2·51-s − 14·53-s + 2·55-s + 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.516·15-s + 0.242·17-s + 0.458·19-s − 0.436·21-s + 0.417·23-s − 4/5·25-s − 0.769·27-s − 0.557·29-s + 1.07·31-s + 0.696·33-s − 0.169·35-s − 0.640·39-s − 1.40·41-s − 1.21·43-s + 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.280·51-s − 1.92·53-s + 0.269·55-s + 0.529·57-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 17 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.27420975596186, −13.76367119613734, −13.50968343423726, −12.93996186658856, −12.35510916973955, −11.83809413604554, −11.37801938660113, −10.73516422087286, −9.895268821910224, −9.722483417037607, −9.384915850599434, −8.690042007070175, −8.233789912267624, −7.792281433330527, −7.152388322763968, −6.565091895848361, −6.171537084409588, −5.202640820437421, −5.065328298111609, −3.920691629867680, −3.660973944362264, −2.995528155665453, −2.405986540624672, −1.856456863896058, −1.088541252059920, 0, 1.088541252059920, 1.856456863896058, 2.405986540624672, 2.995528155665453, 3.660973944362264, 3.920691629867680, 5.065328298111609, 5.202640820437421, 6.171537084409588, 6.565091895848361, 7.152388322763968, 7.792281433330527, 8.233789912267624, 8.690042007070175, 9.384915850599434, 9.722483417037607, 9.895268821910224, 10.73516422087286, 11.37801938660113, 11.83809413604554, 12.35510916973955, 12.93996186658856, 13.50968343423726, 13.76367119613734, 14.27420975596186

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