Properties

Label 2-76-1.1-c7-0-5
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  − 84.8·3-s + 72.4·5-s − 210.·7-s + 5.02e3·9-s + 2.35e3·11-s + 2.21e3·13-s − 6.15e3·15-s + 2.24e4·17-s − 6.85e3·19-s + 1.78e4·21-s − 3.38e4·23-s − 7.28e4·25-s − 2.40e5·27-s − 9.36e4·29-s − 2.22e5·31-s − 1.99e5·33-s − 1.52e4·35-s + 2.28e5·37-s − 1.88e5·39-s + 6.73e4·41-s + 1.62e5·43-s + 3.63e5·45-s + 8.16e4·47-s − 7.79e5·49-s − 1.90e6·51-s + 9.05e5·53-s + 1.70e5·55-s + ⋯
L(s)  = 1  − 1.81·3-s + 0.259·5-s − 0.231·7-s + 2.29·9-s + 0.532·11-s + 0.280·13-s − 0.470·15-s + 1.10·17-s − 0.229·19-s + 0.420·21-s − 0.579·23-s − 0.932·25-s − 2.35·27-s − 0.713·29-s − 1.34·31-s − 0.966·33-s − 0.0600·35-s + 0.742·37-s − 0.508·39-s + 0.152·41-s + 0.312·43-s + 0.595·45-s + 0.114·47-s − 0.946·49-s − 2.00·51-s + 0.835·53-s + 0.138·55-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 \( 1 + 84.8T + 2.18e3T^{2} \)
5 \( 1 - 72.4T + 7.81e4T^{2} \)
7 \( 1 + 210.T + 8.23e5T^{2} \)
11 \( 1 - 2.35e3T + 1.94e7T^{2} \)
13 \( 1 - 2.21e3T + 6.27e7T^{2} \)
17 \( 1 - 2.24e4T + 4.10e8T^{2} \)
23 \( 1 + 3.38e4T + 3.40e9T^{2} \)
29 \( 1 + 9.36e4T + 1.72e10T^{2} \)
31 \( 1 + 2.22e5T + 2.75e10T^{2} \)
37 \( 1 - 2.28e5T + 9.49e10T^{2} \)
41 \( 1 - 6.73e4T + 1.94e11T^{2} \)
43 \( 1 - 1.62e5T + 2.71e11T^{2} \)
47 \( 1 - 8.16e4T + 5.06e11T^{2} \)
53 \( 1 - 9.05e5T + 1.17e12T^{2} \)
59 \( 1 + 1.42e6T + 2.48e12T^{2} \)
61 \( 1 - 2.18e6T + 3.14e12T^{2} \)
67 \( 1 + 4.73e6T + 6.06e12T^{2} \)
71 \( 1 + 3.58e6T + 9.09e12T^{2} \)
73 \( 1 + 3.22e6T + 1.10e13T^{2} \)
79 \( 1 - 3.34e5T + 1.92e13T^{2} \)
83 \( 1 - 9.68e6T + 2.71e13T^{2} \)
89 \( 1 + 6.74e6T + 4.42e13T^{2} \)
97 \( 1 + 4.83e5T + 8.07e13T^{2} \)
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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

−12.28152565358150837347908379420, −11.48693114471259819624152510227, −10.47137730115121103160424066662, −9.486121334147632421540223680204, −7.49401862123509718419737984826, −6.21747090570803598503663720066, −5.50022837011900447565352460731, −4.00684480272684649751168576585, −1.43389284203445057390027421117, 0, 1.43389284203445057390027421117, 4.00684480272684649751168576585, 5.50022837011900447565352460731, 6.21747090570803598503663720066, 7.49401862123509718419737984826, 9.486121334147632421540223680204, 10.47137730115121103160424066662, 11.48693114471259819624152510227, 12.28152565358150837347908379420

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