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Properties

Label 2-76-19.12-c2-0-2
Degree $2$
Conductor $76$
Sign $0.943 + 0.331i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.72 − 2.72i)3-s + (3.36 + 5.82i)5-s − 10.3·7-s + (10.3 − 17.9i)9-s − 11.0·11-s + (2.58 + 1.49i)13-s + (31.8 + 18.3i)15-s + (3.79 + 6.57i)17-s + (−13.7 − 13.1i)19-s + (−48.7 + 28.1i)21-s + (6.15 − 10.6i)23-s + (−10.1 + 17.5i)25-s − 64.2i·27-s + (0.256 + 0.148i)29-s + 43.6i·31-s + ⋯
L(s)  = 1  + (1.57 − 0.909i)3-s + (0.672 + 1.16i)5-s − 1.47·7-s + (1.15 − 1.99i)9-s − 1.00·11-s + (0.198 + 0.114i)13-s + (2.12 + 1.22i)15-s + (0.223 + 0.386i)17-s + (−0.722 − 0.691i)19-s + (−2.32 + 1.34i)21-s + (0.267 − 0.463i)23-s + (−0.405 + 0.702i)25-s − 2.37i·27-s + (0.00885 + 0.00511i)29-s + 1.40i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.943 + 0.331i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.943 + 0.331i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.80762 - 0.308305i\)
\(L(\frac12)\) \(\approx\) \(1.80762 - 0.308305i\)
\(L(2)\) 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 \( 1 + (13.7 + 13.1i)T \)
good3 \( 1 + (-4.72 + 2.72i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.36 - 5.82i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + 11.0T + 121T^{2} \)
13 \( 1 + (-2.58 - 1.49i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-3.79 - 6.57i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-6.15 + 10.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.256 - 0.148i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 43.6iT - 961T^{2} \)
37 \( 1 - 36.3iT - 1.36e3T^{2} \)
41 \( 1 + (-21.8 + 12.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-25.2 - 43.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-31.4 + 54.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (49.5 + 28.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-53.8 + 31.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-17.4 + 30.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-27.8 - 16.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-12.9 + 7.48i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (30.6 + 53.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-0.933 + 0.539i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 128.T + 6.88e3T^{2} \)
89 \( 1 + (-73.8 - 42.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-24.9 + 14.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−14.08315884178417431823421474545, −13.21817696110369345152400287593, −12.65619642373934533855501894221, −10.53937124673195685410425051494, −9.620142466830463681176331269554, −8.450481085917630222352630983483, −7.05619984472060296593726808672, −6.39261298251896814817078344319, −3.27512026723390197075755894112, −2.46907859416641074478735970316, 2.60344106123174269171935662611, 4.05174796596442762145173684898, 5.62104291804695636439714492236, 7.75677759021021468197331778447, 8.965104347800521342173225635543, 9.562961733878902979642964262972, 10.41041581484193727572332861615, 12.83366391472540583499308788221, 13.17506155912334526315424768428, 14.23448964052447396347424668197

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