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Properties

Label 2-76-19.18-c4-0-4
Degree $2$
Conductor $76$
Sign $-0.0200 + 0.999i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12.5i·3-s + 27.8·5-s + 50.6·7-s − 75.8·9-s − 26.1·11-s − 285. i·13-s − 348. i·15-s − 447.·17-s + (−7.24 + 360. i)19-s − 634. i·21-s + 830.·23-s + 149.·25-s − 64.8i·27-s − 338. i·29-s + 669. i·31-s + ⋯
L(s)  = 1  − 1.39i·3-s + 1.11·5-s + 1.03·7-s − 0.936·9-s − 0.216·11-s − 1.69i·13-s − 1.54i·15-s − 1.54·17-s + (−0.0200 + 0.999i)19-s − 1.43i·21-s + 1.57·23-s + 0.238·25-s − 0.0889i·27-s − 0.402i·29-s + 0.696i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0200 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0200 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0200 + 0.999i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.0200 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.42746 - 1.45642i\)
\(L(\frac12)\) \(\approx\) \(1.42746 - 1.45642i\)
\(L(3)\) 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 \( 1 + (7.24 - 360. i)T \)
good3 \( 1 + 12.5iT - 81T^{2} \)
5 \( 1 - 27.8T + 625T^{2} \)
7 \( 1 - 50.6T + 2.40e3T^{2} \)
11 \( 1 + 26.1T + 1.46e4T^{2} \)
13 \( 1 + 285. iT - 2.85e4T^{2} \)
17 \( 1 + 447.T + 8.35e4T^{2} \)
23 \( 1 - 830.T + 2.79e5T^{2} \)
29 \( 1 + 338. iT - 7.07e5T^{2} \)
31 \( 1 - 669. iT - 9.23e5T^{2} \)
37 \( 1 - 623. iT - 1.87e6T^{2} \)
41 \( 1 - 1.72e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.36e3T + 3.41e6T^{2} \)
47 \( 1 + 3.46T + 4.87e6T^{2} \)
53 \( 1 + 1.17e3iT - 7.89e6T^{2} \)
59 \( 1 - 4.43e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.91e3T + 1.38e7T^{2} \)
67 \( 1 + 3.67e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.27e3iT - 2.54e7T^{2} \)
73 \( 1 - 616.T + 2.83e7T^{2} \)
79 \( 1 + 674. iT - 3.89e7T^{2} \)
83 \( 1 - 5.47e3T + 4.74e7T^{2} \)
89 \( 1 - 3.95e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.28e4iT - 8.85e7T^{2} \)
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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

−13.24514201146621742770946701525, −12.81739855254292719961605438714, −11.37698286568323546172040703040, −10.25345816494240752580607807039, −8.609164309424950162694470031989, −7.63490791556213587797630919962, −6.35103751087312107072356451961, −5.19295307248974248412003812756, −2.44339084635702017117481347174, −1.18705625311142663800885977264, 2.14086081686220394619068835180, 4.33404689932454858259637063354, 5.16844690610362049529476660827, 6.81947731810875698559192978406, 8.936113219301257894994858416627, 9.368734127498237145421044773280, 10.79270671292921967456998147279, 11.33354269446986374621154256463, 13.24963959073954136969737794008, 14.19390803069603619193101661784

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