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Properties

Label 2-76-19.18-c10-0-12
Degree $2$
Conductor $76$
Sign $0.275 + 0.961i$
Analytic cond. $48.2871$
Root an. cond. $6.94889$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 142. i·3-s + 3.67e3·5-s − 1.34e4·7-s + 3.87e4·9-s − 1.00e5·11-s − 6.20e5i·13-s + 5.23e5i·15-s + 6.98e5·17-s + (−6.82e5 − 2.38e6i)19-s − 1.91e6i·21-s − 1.05e7·23-s + 3.73e6·25-s + 1.39e7i·27-s + 3.25e6i·29-s + 2.10e7i·31-s + ⋯
L(s)  = 1  + 0.586i·3-s + 1.17·5-s − 0.799·7-s + 0.655·9-s − 0.625·11-s − 1.67i·13-s + 0.689i·15-s + 0.491·17-s + (−0.275 − 0.961i)19-s − 0.469i·21-s − 1.63·23-s + 0.382·25-s + 0.971i·27-s + 0.158i·29-s + 0.735i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.275 + 0.961i$
Analytic conductor: \(48.2871\)
Root analytic conductor: \(6.94889\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5),\ 0.275 + 0.961i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.35973 - 1.02475i\)
\(L(\frac12)\) \(\approx\) \(1.35973 - 1.02475i\)
\(L(6)\) 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 \( 1 + (6.82e5 + 2.38e6i)T \)
good3 \( 1 - 142. iT - 5.90e4T^{2} \)
5 \( 1 - 3.67e3T + 9.76e6T^{2} \)
7 \( 1 + 1.34e4T + 2.82e8T^{2} \)
11 \( 1 + 1.00e5T + 2.59e10T^{2} \)
13 \( 1 + 6.20e5iT - 1.37e11T^{2} \)
17 \( 1 - 6.98e5T + 2.01e12T^{2} \)
23 \( 1 + 1.05e7T + 4.14e13T^{2} \)
29 \( 1 - 3.25e6iT - 4.20e14T^{2} \)
31 \( 1 - 2.10e7iT - 8.19e14T^{2} \)
37 \( 1 + 7.96e7iT - 4.80e15T^{2} \)
41 \( 1 + 2.25e8iT - 1.34e16T^{2} \)
43 \( 1 - 2.27e8T + 2.16e16T^{2} \)
47 \( 1 + 1.47e8T + 5.25e16T^{2} \)
53 \( 1 - 1.13e7iT - 1.74e17T^{2} \)
59 \( 1 + 4.79e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.12e9T + 7.13e17T^{2} \)
67 \( 1 + 1.88e9iT - 1.82e18T^{2} \)
71 \( 1 - 1.27e8iT - 3.25e18T^{2} \)
73 \( 1 + 1.59e9T + 4.29e18T^{2} \)
79 \( 1 + 5.62e9iT - 9.46e18T^{2} \)
83 \( 1 - 4.12e9T + 1.55e19T^{2} \)
89 \( 1 + 6.15e9iT - 3.11e19T^{2} \)
97 \( 1 - 7.20e9iT - 7.37e19T^{2} \)
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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

−12.56414269951808147019279920335, −10.53237951764582327960895402935, −10.14457264858417607836716631693, −9.143544452538667227493968964412, −7.56950927769981993692476942465, −6.08365484823716886752008360383, −5.16415994253635547096398352312, −3.51378201353323001652066575720, −2.20130338157584104974750471913, −0.43625907917168895252593527557, 1.42038253796626696364186072926, 2.34769993301372960493394169668, 4.16895901854917403928595377002, 5.91968661426238655734241436557, 6.63595689149884262315859295624, 7.994670645203421118522640916644, 9.658766900435843519449027278098, 10.02342687473025061395496382357, 11.77679175209606793264315082768, 12.84537313536266988612949800979

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