Properties

Label 2-76-1.1-c5-0-3
Degree $2$
Conductor $76$
Sign $1$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17.5·3-s + 87.4·5-s + 76.9·7-s + 64.9·9-s − 99.2·11-s − 79.5·13-s + 1.53e3·15-s − 465.·17-s − 361·19-s + 1.34e3·21-s + 2.23e3·23-s + 4.52e3·25-s − 3.12e3·27-s + 757.·29-s + 2.71e3·31-s − 1.74e3·33-s + 6.72e3·35-s + 1.03e4·37-s − 1.39e3·39-s − 6.94e3·41-s − 2.10e3·43-s + 5.68e3·45-s − 2.05e4·47-s − 1.08e4·49-s − 8.17e3·51-s − 1.75e4·53-s − 8.67e3·55-s + ⋯
L(s)  = 1  + 1.12·3-s + 1.56·5-s + 0.593·7-s + 0.267·9-s − 0.247·11-s − 0.130·13-s + 1.76·15-s − 0.391·17-s − 0.229·19-s + 0.668·21-s + 0.879·23-s + 1.44·25-s − 0.824·27-s + 0.167·29-s + 0.507·31-s − 0.278·33-s + 0.928·35-s + 1.23·37-s − 0.146·39-s − 0.645·41-s − 0.173·43-s + 0.418·45-s − 1.35·47-s − 0.647·49-s − 0.440·51-s − 0.859·53-s − 0.386·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.232719759\)
\(L(\frac12)\) \(\approx\) \(3.232719759\)
\(L(\frac{7}{2})\) 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 \( 1 + 361T \)
good3 \( 1 - 17.5T + 243T^{2} \)
5 \( 1 - 87.4T + 3.12e3T^{2} \)
7 \( 1 - 76.9T + 1.68e4T^{2} \)
11 \( 1 + 99.2T + 1.61e5T^{2} \)
13 \( 1 + 79.5T + 3.71e5T^{2} \)
17 \( 1 + 465.T + 1.41e6T^{2} \)
23 \( 1 - 2.23e3T + 6.43e6T^{2} \)
29 \( 1 - 757.T + 2.05e7T^{2} \)
31 \( 1 - 2.71e3T + 2.86e7T^{2} \)
37 \( 1 - 1.03e4T + 6.93e7T^{2} \)
41 \( 1 + 6.94e3T + 1.15e8T^{2} \)
43 \( 1 + 2.10e3T + 1.47e8T^{2} \)
47 \( 1 + 2.05e4T + 2.29e8T^{2} \)
53 \( 1 + 1.75e4T + 4.18e8T^{2} \)
59 \( 1 + 3.45e4T + 7.14e8T^{2} \)
61 \( 1 + 4.09e4T + 8.44e8T^{2} \)
67 \( 1 - 6.82e4T + 1.35e9T^{2} \)
71 \( 1 - 3.88e4T + 1.80e9T^{2} \)
73 \( 1 + 8.36e4T + 2.07e9T^{2} \)
79 \( 1 - 3.90e4T + 3.07e9T^{2} \)
83 \( 1 - 9.94e4T + 3.93e9T^{2} \)
89 \( 1 + 8.89e4T + 5.58e9T^{2} \)
97 \( 1 + 8.74e4T + 8.58e9T^{2} \)
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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

−13.67976121231502930489182554393, −12.89518923131157052972504820759, −11.14761601619831445480702423221, −9.880041180394734203892525922465, −9.039202755430444181666244516576, −7.972373649925004474354322312758, −6.35828208620470384198163370998, −4.93379466306303632770095227429, −2.87019025177418279132846485714, −1.74145827011482975363307579674, 1.74145827011482975363307579674, 2.87019025177418279132846485714, 4.93379466306303632770095227429, 6.35828208620470384198163370998, 7.972373649925004474354322312758, 9.039202755430444181666244516576, 9.880041180394734203892525922465, 11.14761601619831445480702423221, 12.89518923131157052972504820759, 13.67976121231502930489182554393

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