Properties

Label 2-76-1.1-c7-0-8
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  + 8.09·3-s + 276.·5-s − 720.·7-s − 2.12e3·9-s − 4.97e3·11-s + 8.08e3·13-s + 2.23e3·15-s + 1.37e3·17-s − 6.85e3·19-s − 5.83e3·21-s − 5.50e4·23-s − 1.76e3·25-s − 3.48e4·27-s − 1.17e5·29-s − 1.51e5·31-s − 4.02e4·33-s − 1.99e5·35-s − 7.60e4·37-s + 6.54e4·39-s + 4.20e5·41-s − 4.11e5·43-s − 5.86e5·45-s − 7.91e5·47-s − 3.03e5·49-s + 1.11e4·51-s − 8.00e5·53-s − 1.37e6·55-s + ⋯
L(s)  = 1  + 0.173·3-s + 0.988·5-s − 0.794·7-s − 0.970·9-s − 1.12·11-s + 1.02·13-s + 0.171·15-s + 0.0679·17-s − 0.229·19-s − 0.137·21-s − 0.943·23-s − 0.0225·25-s − 0.340·27-s − 0.896·29-s − 0.910·31-s − 0.194·33-s − 0.785·35-s − 0.246·37-s + 0.176·39-s + 0.953·41-s − 0.789·43-s − 0.959·45-s − 1.11·47-s − 0.369·49-s + 0.0117·51-s − 0.738·53-s − 1.11·55-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 \( 1 - 8.09T + 2.18e3T^{2} \)
5 \( 1 - 276.T + 7.81e4T^{2} \)
7 \( 1 + 720.T + 8.23e5T^{2} \)
11 \( 1 + 4.97e3T + 1.94e7T^{2} \)
13 \( 1 - 8.08e3T + 6.27e7T^{2} \)
17 \( 1 - 1.37e3T + 4.10e8T^{2} \)
23 \( 1 + 5.50e4T + 3.40e9T^{2} \)
29 \( 1 + 1.17e5T + 1.72e10T^{2} \)
31 \( 1 + 1.51e5T + 2.75e10T^{2} \)
37 \( 1 + 7.60e4T + 9.49e10T^{2} \)
41 \( 1 - 4.20e5T + 1.94e11T^{2} \)
43 \( 1 + 4.11e5T + 2.71e11T^{2} \)
47 \( 1 + 7.91e5T + 5.06e11T^{2} \)
53 \( 1 + 8.00e5T + 1.17e12T^{2} \)
59 \( 1 - 3.09e6T + 2.48e12T^{2} \)
61 \( 1 + 2.19e6T + 3.14e12T^{2} \)
67 \( 1 + 4.20e5T + 6.06e12T^{2} \)
71 \( 1 - 2.69e6T + 9.09e12T^{2} \)
73 \( 1 - 5.98e5T + 1.10e13T^{2} \)
79 \( 1 - 2.62e6T + 1.92e13T^{2} \)
83 \( 1 + 5.86e6T + 2.71e13T^{2} \)
89 \( 1 - 6.26e6T + 4.42e13T^{2} \)
97 \( 1 - 1.21e7T + 8.07e13T^{2} \)
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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

−12.86218786378153143206425250425, −11.28606520353128882358194508217, −10.19029757647248984516559447459, −9.183490035641235671428248284628, −7.998995673142444779296141374813, −6.28562746600711123647841210183, −5.48429504003763877122652506159, −3.40615805873004771621698979568, −2.08106368941208712746800657076, 0, 2.08106368941208712746800657076, 3.40615805873004771621698979568, 5.48429504003763877122652506159, 6.28562746600711123647841210183, 7.998995673142444779296141374813, 9.183490035641235671428248284628, 10.19029757647248984516559447459, 11.28606520353128882358194508217, 12.86218786378153143206425250425

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