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Properties

Label 2-76664-1.1-c1-0-7
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  + 5-s − 7-s − 3·9-s + 5·11-s + 13-s − 6·17-s + 4·19-s − 4·25-s − 5·31-s − 35-s + 2·41-s − 2·43-s − 3·45-s − 2·47-s + 49-s + 9·53-s + 5·55-s − 3·59-s − 6·61-s + 3·63-s + 65-s + 5·67-s + 71-s + 10·73-s − 5·77-s − 4·79-s + 9·81-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 9-s + 1.50·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 4/5·25-s − 0.898·31-s − 0.169·35-s + 0.312·41-s − 0.304·43-s − 0.447·45-s − 0.291·47-s + 1/7·49-s + 1.23·53-s + 0.674·55-s − 0.390·59-s − 0.768·61-s + 0.377·63-s + 0.124·65-s + 0.610·67-s + 0.118·71-s + 1.17·73-s − 0.569·77-s − 0.450·79-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: Trivial
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 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−14.19123318018255, −13.85411919570975, −13.38359591186764, −12.90372212403776, −12.16966838994404, −11.77205673402695, −11.32678287892718, −10.93113085823963, −10.26608772433643, −9.560568369138143, −9.203819722867994, −8.932449213541405, −8.285164884384055, −7.664605169286221, −6.890681589635921, −6.583976941878684, −6.001839292724342, −5.575378646466739, −4.917559740171179, −4.116464787839076, −3.696922538993162, −3.064402135031209, −2.299109014555241, −1.753204922658817, −0.9049240313267710, 0, 0.9049240313267710, 1.753204922658817, 2.299109014555241, 3.064402135031209, 3.696922538993162, 4.116464787839076, 4.917559740171179, 5.575378646466739, 6.001839292724342, 6.583976941878684, 6.890681589635921, 7.664605169286221, 8.285164884384055, 8.932449213541405, 9.203819722867994, 9.560568369138143, 10.26608772433643, 10.93113085823963, 11.32678287892718, 11.77205673402695, 12.16966838994404, 12.90372212403776, 13.38359591186764, 13.85411919570975, 14.19123318018255

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