Properties

Label 2-76-1.1-c7-0-7
Degree $2$
Conductor $76$
Sign $-1$
Analytic cond. $23.7412$
Root an. cond. $4.87250$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 36.5·3-s − 266.·5-s + 1.62e3·7-s − 853.·9-s + 3.31e3·11-s + 6.65e3·13-s + 9.73e3·15-s − 3.05e4·17-s − 6.85e3·19-s − 5.94e4·21-s − 2.94e4·23-s − 7.09e3·25-s + 1.11e5·27-s − 1.19e5·29-s + 6.72e3·31-s − 1.20e5·33-s − 4.33e5·35-s − 2.97e5·37-s − 2.43e5·39-s − 1.11e5·41-s − 4.51e5·43-s + 2.27e5·45-s − 1.04e6·47-s + 1.82e6·49-s + 1.11e6·51-s + 2.48e4·53-s − 8.82e5·55-s + ⋯
L(s)  = 1  − 0.780·3-s − 0.953·5-s + 1.79·7-s − 0.390·9-s + 0.750·11-s + 0.840·13-s + 0.744·15-s − 1.50·17-s − 0.229·19-s − 1.40·21-s − 0.505·23-s − 0.0908·25-s + 1.08·27-s − 0.913·29-s + 0.0405·31-s − 0.585·33-s − 1.71·35-s − 0.966·37-s − 0.656·39-s − 0.253·41-s − 0.866·43-s + 0.372·45-s − 1.46·47-s + 2.21·49-s + 1.17·51-s + 0.0228·53-s − 0.715·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(23.7412\)
Root analytic conductor: \(4.87250\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + 6.85e3T \)
good3 \( 1 + 36.5T + 2.18e3T^{2} \)
5 \( 1 + 266.T + 7.81e4T^{2} \)
7 \( 1 - 1.62e3T + 8.23e5T^{2} \)
11 \( 1 - 3.31e3T + 1.94e7T^{2} \)
13 \( 1 - 6.65e3T + 6.27e7T^{2} \)
17 \( 1 + 3.05e4T + 4.10e8T^{2} \)
23 \( 1 + 2.94e4T + 3.40e9T^{2} \)
29 \( 1 + 1.19e5T + 1.72e10T^{2} \)
31 \( 1 - 6.72e3T + 2.75e10T^{2} \)
37 \( 1 + 2.97e5T + 9.49e10T^{2} \)
41 \( 1 + 1.11e5T + 1.94e11T^{2} \)
43 \( 1 + 4.51e5T + 2.71e11T^{2} \)
47 \( 1 + 1.04e6T + 5.06e11T^{2} \)
53 \( 1 - 2.48e4T + 1.17e12T^{2} \)
59 \( 1 + 1.72e6T + 2.48e12T^{2} \)
61 \( 1 + 1.66e6T + 3.14e12T^{2} \)
67 \( 1 - 2.99e6T + 6.06e12T^{2} \)
71 \( 1 + 2.90e6T + 9.09e12T^{2} \)
73 \( 1 - 4.36e6T + 1.10e13T^{2} \)
79 \( 1 - 3.76e6T + 1.92e13T^{2} \)
83 \( 1 + 1.62e6T + 2.71e13T^{2} \)
89 \( 1 - 2.25e6T + 4.42e13T^{2} \)
97 \( 1 + 1.60e7T + 8.07e13T^{2} \)
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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

−12.05203909351585932786985895068, −11.34045784306243670628369701632, −10.92160535217199050822724887666, −8.786913459364789843395027865195, −7.982076131365813559284209642941, −6.50526427399407943655478232706, −5.05865578111671347205654370860, −3.98567431597430609171657104326, −1.64709687339730579241851273778, 0, 1.64709687339730579241851273778, 3.98567431597430609171657104326, 5.05865578111671347205654370860, 6.50526427399407943655478232706, 7.982076131365813559284209642941, 8.786913459364789843395027865195, 10.92160535217199050822724887666, 11.34045784306243670628369701632, 12.05203909351585932786985895068

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