Properties

Label 2-76-1.1-c5-0-1
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  − 9.77·3-s − 24.1·5-s + 64.5·7-s − 147.·9-s − 58.5·11-s + 875.·13-s + 235.·15-s + 2.05e3·17-s − 361·19-s − 630.·21-s + 4.90e3·23-s − 2.54e3·25-s + 3.81e3·27-s + 1.70e3·29-s + 3.65e3·31-s + 572.·33-s − 1.55e3·35-s − 2.20e3·37-s − 8.55e3·39-s − 1.22e4·41-s + 1.09e4·43-s + 3.56e3·45-s + 1.21e4·47-s − 1.26e4·49-s − 2.00e4·51-s − 9.12e3·53-s + 1.41e3·55-s + ⋯
L(s)  = 1  − 0.626·3-s − 0.431·5-s + 0.498·7-s − 0.607·9-s − 0.146·11-s + 1.43·13-s + 0.270·15-s + 1.72·17-s − 0.229·19-s − 0.312·21-s + 1.93·23-s − 0.813·25-s + 1.00·27-s + 0.376·29-s + 0.682·31-s + 0.0915·33-s − 0.215·35-s − 0.264·37-s − 0.900·39-s − 1.14·41-s + 0.900·43-s + 0.262·45-s + 0.799·47-s − 0.751·49-s − 1.07·51-s − 0.446·53-s + 0.0630·55-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

\(L(3)\) \(\approx\) \(1.382736258\)
\(L(\frac12)\) \(\approx\) \(1.382736258\)
\(L(\frac{7}{2})\) 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 \( 1 + 9.77T + 243T^{2} \)
5 \( 1 + 24.1T + 3.12e3T^{2} \)
7 \( 1 - 64.5T + 1.68e4T^{2} \)
11 \( 1 + 58.5T + 1.61e5T^{2} \)
13 \( 1 - 875.T + 3.71e5T^{2} \)
17 \( 1 - 2.05e3T + 1.41e6T^{2} \)
23 \( 1 - 4.90e3T + 6.43e6T^{2} \)
29 \( 1 - 1.70e3T + 2.05e7T^{2} \)
31 \( 1 - 3.65e3T + 2.86e7T^{2} \)
37 \( 1 + 2.20e3T + 6.93e7T^{2} \)
41 \( 1 + 1.22e4T + 1.15e8T^{2} \)
43 \( 1 - 1.09e4T + 1.47e8T^{2} \)
47 \( 1 - 1.21e4T + 2.29e8T^{2} \)
53 \( 1 + 9.12e3T + 4.18e8T^{2} \)
59 \( 1 - 1.28e4T + 7.14e8T^{2} \)
61 \( 1 + 1.31e4T + 8.44e8T^{2} \)
67 \( 1 + 6.29e4T + 1.35e9T^{2} \)
71 \( 1 - 2.26e4T + 1.80e9T^{2} \)
73 \( 1 - 7.05e3T + 2.07e9T^{2} \)
79 \( 1 - 7.74e4T + 3.07e9T^{2} \)
83 \( 1 - 1.33e4T + 3.93e9T^{2} \)
89 \( 1 - 1.25e5T + 5.58e9T^{2} \)
97 \( 1 - 1.16e5T + 8.58e9T^{2} \)
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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

−13.53397523702197376272851810613, −12.19084503281156914079856152190, −11.35277163568279244322129940486, −10.49669785411483836857834461260, −8.824502654686296245969860278778, −7.77926596899161802417837212272, −6.20547181813379291535045415702, −5.06819908938554341504096715126, −3.34583163915862463514125538093, −0.983451502948084401135874345507, 0.983451502948084401135874345507, 3.34583163915862463514125538093, 5.06819908938554341504096715126, 6.20547181813379291535045415702, 7.77926596899161802417837212272, 8.824502654686296245969860278778, 10.49669785411483836857834461260, 11.35277163568279244322129940486, 12.19084503281156914079856152190, 13.53397523702197376272851810613

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