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Properties

Label 2-76-19.12-c4-0-1
Degree $2$
Conductor $76$
Sign $-0.663 + 0.748i$
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  + (−13.1 + 7.59i)3-s + (19.9 + 34.5i)5-s − 53.3·7-s + (74.8 − 129. i)9-s + 9.01·11-s + (−5.39 − 3.11i)13-s + (−524. − 302. i)15-s + (−261. − 453. i)17-s + (−329. + 147. i)19-s + (701. − 405. i)21-s + (204. − 353. i)23-s + (−481. + 834. i)25-s + 1.04e3i·27-s + (1.17e3 + 679. i)29-s − 214. i·31-s + ⋯
L(s)  = 1  + (−1.46 + 0.843i)3-s + (0.797 + 1.38i)5-s − 1.08·7-s + (0.923 − 1.59i)9-s + 0.0744·11-s + (−0.0319 − 0.0184i)13-s + (−2.32 − 1.34i)15-s + (−0.905 − 1.56i)17-s + (−0.913 + 0.407i)19-s + (1.59 − 0.918i)21-s + (0.385 − 0.668i)23-s + (−0.770 + 1.33i)25-s + 1.42i·27-s + (1.39 + 0.807i)29-s − 0.223i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.663 + 0.748i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.663 + 0.748i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0854082 - 0.189773i\)
\(L(\frac12)\) \(\approx\) \(0.0854082 - 0.189773i\)
\(L(3)\) 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 \( 1 + (329. - 147. i)T \)
good3 \( 1 + (13.1 - 7.59i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-19.9 - 34.5i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 53.3T + 2.40e3T^{2} \)
11 \( 1 - 9.01T + 1.46e4T^{2} \)
13 \( 1 + (5.39 + 3.11i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (261. + 453. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-204. + 353. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.17e3 - 679. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 214. iT - 9.23e5T^{2} \)
37 \( 1 - 1.22e3iT - 1.87e6T^{2} \)
41 \( 1 + (666. - 384. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.14e3 + 1.98e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.74e3 - 3.01e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (4.81e3 + 2.78e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (2.13e3 - 1.23e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.47e3 - 2.54e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.47e3 - 851. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (2.98e3 - 1.72e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (4.12e3 + 7.13e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-7.35e3 + 4.24e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 2.75e3T + 4.74e7T^{2} \)
89 \( 1 + (-5.98e3 - 3.45e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (1.05e4 - 6.07e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−14.59689858427891823211034892181, −13.33899835154470546082960188114, −12.00221141985894554390007342674, −10.88237294825744514759415633169, −10.26079823910724971047865384358, −9.390111498669414888700672989041, −6.64049174950814692596212586729, −6.39939945224618811308929567673, −4.83136843203201552588995301267, −2.99930136862071314558739584448, 0.12515793293169903243008735216, 1.62054595554836339707053918007, 4.68120545399771299612461343940, 5.97147813575086799187138999123, 6.59499199767520350070577559274, 8.471521467673060574588886942938, 9.777862292482920408860725444361, 10.98576962728199557312573536235, 12.36885184769041962464781841927, 12.85741818362960343520632632943

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