Properties

Label 2-76-1.1-c5-0-0
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  − 25.7·3-s − 109.·5-s − 177.·7-s + 420.·9-s + 228.·11-s − 476.·13-s + 2.81e3·15-s − 1.86e3·17-s − 361·19-s + 4.56e3·21-s + 1.96e3·23-s + 8.80e3·25-s − 4.57e3·27-s − 4.20e3·29-s − 3.84e3·31-s − 5.87e3·33-s + 1.93e4·35-s − 1.54e4·37-s + 1.22e4·39-s + 1.92e3·41-s − 7.33e3·43-s − 4.59e4·45-s + 5.08e3·47-s + 1.46e4·49-s + 4.81e4·51-s − 1.26e4·53-s − 2.49e4·55-s + ⋯
L(s)  = 1  − 1.65·3-s − 1.95·5-s − 1.36·7-s + 1.73·9-s + 0.568·11-s − 0.781·13-s + 3.22·15-s − 1.56·17-s − 0.229·19-s + 2.26·21-s + 0.774·23-s + 2.81·25-s − 1.20·27-s − 0.927·29-s − 0.718·31-s − 0.938·33-s + 2.67·35-s − 1.85·37-s + 1.29·39-s + 0.179·41-s − 0.604·43-s − 3.38·45-s + 0.335·47-s + 0.870·49-s + 2.59·51-s − 0.616·53-s − 1.11·55-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

\(L(3)\) \(\approx\) \(0.1211080215\)
\(L(\frac12)\) \(\approx\) \(0.1211080215\)
\(L(\frac{7}{2})\) 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 \( 1 + 25.7T + 243T^{2} \)
5 \( 1 + 109.T + 3.12e3T^{2} \)
7 \( 1 + 177.T + 1.68e4T^{2} \)
11 \( 1 - 228.T + 1.61e5T^{2} \)
13 \( 1 + 476.T + 3.71e5T^{2} \)
17 \( 1 + 1.86e3T + 1.41e6T^{2} \)
23 \( 1 - 1.96e3T + 6.43e6T^{2} \)
29 \( 1 + 4.20e3T + 2.05e7T^{2} \)
31 \( 1 + 3.84e3T + 2.86e7T^{2} \)
37 \( 1 + 1.54e4T + 6.93e7T^{2} \)
41 \( 1 - 1.92e3T + 1.15e8T^{2} \)
43 \( 1 + 7.33e3T + 1.47e8T^{2} \)
47 \( 1 - 5.08e3T + 2.29e8T^{2} \)
53 \( 1 + 1.26e4T + 4.18e8T^{2} \)
59 \( 1 - 7.71e3T + 7.14e8T^{2} \)
61 \( 1 + 1.79e4T + 8.44e8T^{2} \)
67 \( 1 - 1.07e4T + 1.35e9T^{2} \)
71 \( 1 - 5.06e4T + 1.80e9T^{2} \)
73 \( 1 + 5.80e4T + 2.07e9T^{2} \)
79 \( 1 - 3.39e4T + 3.07e9T^{2} \)
83 \( 1 - 3.07e4T + 3.93e9T^{2} \)
89 \( 1 - 9.76e4T + 5.58e9T^{2} \)
97 \( 1 - 1.05e5T + 8.58e9T^{2} \)
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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

−12.92494875313522705827480024417, −12.25407541420771425735302522872, −11.42560098733300591491520879725, −10.63518104743417938399057745356, −9.050439576922314909830268289978, −7.20521179131213236927060556530, −6.58451447593414980970943941913, −4.85605074952776546825579491217, −3.67637079863556572563571874213, −0.27305029903724779261781600937, 0.27305029903724779261781600937, 3.67637079863556572563571874213, 4.85605074952776546825579491217, 6.58451447593414980970943941913, 7.20521179131213236927060556530, 9.050439576922314909830268289978, 10.63518104743417938399057745356, 11.42560098733300591491520879725, 12.25407541420771425735302522872, 12.92494875313522705827480024417

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