Properties

Label 2-76-1.1-c5-0-2
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  + 27.9·3-s − 64.0·5-s + 65.8·7-s + 539.·9-s + 635.·11-s + 467.·13-s − 1.79e3·15-s + 522.·17-s − 361·19-s + 1.84e3·21-s − 3.22e3·23-s + 981.·25-s + 8.30e3·27-s + 6.97e3·29-s − 3.38e3·31-s + 1.77e4·33-s − 4.21e3·35-s − 1.34e4·37-s + 1.30e4·39-s + 7.10e3·41-s − 1.40e4·43-s − 3.46e4·45-s − 1.44e3·47-s − 1.24e4·49-s + 1.46e4·51-s − 3.71e4·53-s − 4.07e4·55-s + ⋯
L(s)  = 1  + 1.79·3-s − 1.14·5-s + 0.507·7-s + 2.22·9-s + 1.58·11-s + 0.767·13-s − 2.05·15-s + 0.438·17-s − 0.229·19-s + 0.911·21-s − 1.26·23-s + 0.313·25-s + 2.19·27-s + 1.54·29-s − 0.633·31-s + 2.84·33-s − 0.581·35-s − 1.61·37-s + 1.37·39-s + 0.660·41-s − 1.15·43-s − 2.54·45-s − 0.0953·47-s − 0.742·49-s + 0.786·51-s − 1.81·53-s − 1.81·55-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

\(L(3)\) \(\approx\) \(3.053833143\)
\(L(\frac12)\) \(\approx\) \(3.053833143\)
\(L(\frac{7}{2})\) 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 \( 1 - 27.9T + 243T^{2} \)
5 \( 1 + 64.0T + 3.12e3T^{2} \)
7 \( 1 - 65.8T + 1.68e4T^{2} \)
11 \( 1 - 635.T + 1.61e5T^{2} \)
13 \( 1 - 467.T + 3.71e5T^{2} \)
17 \( 1 - 522.T + 1.41e6T^{2} \)
23 \( 1 + 3.22e3T + 6.43e6T^{2} \)
29 \( 1 - 6.97e3T + 2.05e7T^{2} \)
31 \( 1 + 3.38e3T + 2.86e7T^{2} \)
37 \( 1 + 1.34e4T + 6.93e7T^{2} \)
41 \( 1 - 7.10e3T + 1.15e8T^{2} \)
43 \( 1 + 1.40e4T + 1.47e8T^{2} \)
47 \( 1 + 1.44e3T + 2.29e8T^{2} \)
53 \( 1 + 3.71e4T + 4.18e8T^{2} \)
59 \( 1 - 3.78e4T + 7.14e8T^{2} \)
61 \( 1 - 3.96e4T + 8.44e8T^{2} \)
67 \( 1 + 1.10e4T + 1.35e9T^{2} \)
71 \( 1 - 9.09e3T + 1.80e9T^{2} \)
73 \( 1 - 2.99e4T + 2.07e9T^{2} \)
79 \( 1 + 3.32e4T + 3.07e9T^{2} \)
83 \( 1 + 5.07e4T + 3.93e9T^{2} \)
89 \( 1 + 1.28e5T + 5.58e9T^{2} \)
97 \( 1 + 1.11e5T + 8.58e9T^{2} \)
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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

−13.99073734783949327876782061188, −12.49292016485261935304211822347, −11.46846883695583408908852622592, −9.862517792953791019186471355687, −8.596283517633568583178472943800, −8.121656609414142602846882040521, −6.84120663752561090423812078603, −4.19606431572491288110441971294, −3.45441998824108242666951653188, −1.56753379389982755416183322392, 1.56753379389982755416183322392, 3.45441998824108242666951653188, 4.19606431572491288110441971294, 6.84120663752561090423812078603, 8.121656609414142602846882040521, 8.596283517633568583178472943800, 9.862517792953791019186471355687, 11.46846883695583408908852622592, 12.49292016485261935304211822347, 13.99073734783949327876782061188

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