Properties

Label 2-76-1.1-c7-0-9
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  + 25.4·3-s − 3.35·5-s − 173.·7-s − 1.54e3·9-s + 3.80e3·11-s − 1.44e4·13-s − 85.3·15-s + 3.29e3·17-s − 6.85e3·19-s − 4.40e3·21-s + 643.·23-s − 7.81e4·25-s − 9.47e4·27-s + 9.37e4·29-s − 1.03e5·31-s + 9.67e4·33-s + 581.·35-s − 3.22e5·37-s − 3.67e5·39-s − 8.43e5·41-s − 2.89e5·43-s + 5.17e3·45-s + 7.72e5·47-s − 7.93e5·49-s + 8.37e4·51-s − 5.96e5·53-s − 1.27e4·55-s + ⋯
L(s)  = 1  + 0.543·3-s − 0.0120·5-s − 0.190·7-s − 0.704·9-s + 0.861·11-s − 1.82·13-s − 0.00652·15-s + 0.162·17-s − 0.229·19-s − 0.103·21-s + 0.0110·23-s − 0.999·25-s − 0.926·27-s + 0.713·29-s − 0.626·31-s + 0.468·33-s + 0.00229·35-s − 1.04·37-s − 0.990·39-s − 1.91·41-s − 0.555·43-s + 0.00845·45-s + 1.08·47-s − 0.963·49-s + 0.0883·51-s − 0.550·53-s − 0.0103·55-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 \( 1 - 25.4T + 2.18e3T^{2} \)
5 \( 1 + 3.35T + 7.81e4T^{2} \)
7 \( 1 + 173.T + 8.23e5T^{2} \)
11 \( 1 - 3.80e3T + 1.94e7T^{2} \)
13 \( 1 + 1.44e4T + 6.27e7T^{2} \)
17 \( 1 - 3.29e3T + 4.10e8T^{2} \)
23 \( 1 - 643.T + 3.40e9T^{2} \)
29 \( 1 - 9.37e4T + 1.72e10T^{2} \)
31 \( 1 + 1.03e5T + 2.75e10T^{2} \)
37 \( 1 + 3.22e5T + 9.49e10T^{2} \)
41 \( 1 + 8.43e5T + 1.94e11T^{2} \)
43 \( 1 + 2.89e5T + 2.71e11T^{2} \)
47 \( 1 - 7.72e5T + 5.06e11T^{2} \)
53 \( 1 + 5.96e5T + 1.17e12T^{2} \)
59 \( 1 + 2.76e6T + 2.48e12T^{2} \)
61 \( 1 - 2.39e6T + 3.14e12T^{2} \)
67 \( 1 - 2.71e6T + 6.06e12T^{2} \)
71 \( 1 - 5.03e6T + 9.09e12T^{2} \)
73 \( 1 - 3.65e6T + 1.10e13T^{2} \)
79 \( 1 + 4.72e6T + 1.92e13T^{2} \)
83 \( 1 - 2.28e6T + 2.71e13T^{2} \)
89 \( 1 - 8.80e6T + 4.42e13T^{2} \)
97 \( 1 - 1.21e7T + 8.07e13T^{2} \)
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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

−12.42940034211847781458027378765, −11.60187665478766010875067194320, −10.05882634498070552142241696657, −9.129141060024862101297747243377, −7.939396058304497815093906973133, −6.67284926102135788254185532033, −5.11573175392536565095971963215, −3.49246247150725732406423722658, −2.10847702965545947670379304022, 0, 2.10847702965545947670379304022, 3.49246247150725732406423722658, 5.11573175392536565095971963215, 6.67284926102135788254185532033, 7.939396058304497815093906973133, 9.129141060024862101297747243377, 10.05882634498070552142241696657, 11.60187665478766010875067194320, 12.42940034211847781458027378765

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