Properties

Label 2-76-1.1-c7-0-10
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  + 73.9·3-s − 358.·5-s − 109.·7-s + 3.27e3·9-s − 7.15e3·11-s − 3.12e3·13-s − 2.65e4·15-s − 2.38e4·17-s − 6.85e3·19-s − 8.11e3·21-s + 5.06e4·23-s + 5.07e4·25-s + 8.03e4·27-s − 1.34e5·29-s + 1.99e5·31-s − 5.28e5·33-s + 3.94e4·35-s − 9.54e4·37-s − 2.30e5·39-s − 4.88e5·41-s + 1.62e5·43-s − 1.17e6·45-s − 8.32e5·47-s − 8.11e5·49-s − 1.76e6·51-s + 9.53e5·53-s + 2.56e6·55-s + ⋯
L(s)  = 1  + 1.58·3-s − 1.28·5-s − 0.120·7-s + 1.49·9-s − 1.62·11-s − 0.394·13-s − 2.02·15-s − 1.17·17-s − 0.229·19-s − 0.191·21-s + 0.867·23-s + 0.649·25-s + 0.785·27-s − 1.02·29-s + 1.20·31-s − 2.56·33-s + 0.155·35-s − 0.309·37-s − 0.623·39-s − 1.10·41-s + 0.312·43-s − 1.92·45-s − 1.16·47-s − 0.985·49-s − 1.86·51-s + 0.879·53-s + 2.08·55-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 \( 1 - 73.9T + 2.18e3T^{2} \)
5 \( 1 + 358.T + 7.81e4T^{2} \)
7 \( 1 + 109.T + 8.23e5T^{2} \)
11 \( 1 + 7.15e3T + 1.94e7T^{2} \)
13 \( 1 + 3.12e3T + 6.27e7T^{2} \)
17 \( 1 + 2.38e4T + 4.10e8T^{2} \)
23 \( 1 - 5.06e4T + 3.40e9T^{2} \)
29 \( 1 + 1.34e5T + 1.72e10T^{2} \)
31 \( 1 - 1.99e5T + 2.75e10T^{2} \)
37 \( 1 + 9.54e4T + 9.49e10T^{2} \)
41 \( 1 + 4.88e5T + 1.94e11T^{2} \)
43 \( 1 - 1.62e5T + 2.71e11T^{2} \)
47 \( 1 + 8.32e5T + 5.06e11T^{2} \)
53 \( 1 - 9.53e5T + 1.17e12T^{2} \)
59 \( 1 - 2.46e6T + 2.48e12T^{2} \)
61 \( 1 - 1.15e6T + 3.14e12T^{2} \)
67 \( 1 + 1.59e6T + 6.06e12T^{2} \)
71 \( 1 + 5.90e6T + 9.09e12T^{2} \)
73 \( 1 + 1.17e6T + 1.10e13T^{2} \)
79 \( 1 - 7.58e6T + 1.92e13T^{2} \)
83 \( 1 - 7.26e6T + 2.71e13T^{2} \)
89 \( 1 + 7.80e6T + 4.42e13T^{2} \)
97 \( 1 + 6.56e6T + 8.07e13T^{2} \)
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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

−12.87645839178033409564207417027, −11.41175334672002421337414684915, −10.13180835275794533788599683687, −8.754923715995438851696599823015, −8.013659569154647037090963357717, −7.14087023433922310854911057027, −4.72945928646603151656852285482, −3.42736479409068320434349872190, −2.36430134544470230845786203419, 0, 2.36430134544470230845786203419, 3.42736479409068320434349872190, 4.72945928646603151656852285482, 7.14087023433922310854911057027, 8.013659569154647037090963357717, 8.754923715995438851696599823015, 10.13180835275794533788599683687, 11.41175334672002421337414684915, 12.87645839178033409564207417027

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