Properties

Label 2-76-19.18-c6-0-7
Degree $2$
Conductor $76$
Sign $0.398 + 0.916i$
Analytic cond. $17.4841$
Root an. cond. $4.18140$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 38.1i·3-s + 102.·5-s + 512.·7-s − 726.·9-s + 2.41e3·11-s + 3.77e3i·13-s − 3.90e3i·15-s + 1.56e3·17-s + (−2.73e3 − 6.28e3i)19-s − 1.95e4i·21-s − 626.·23-s − 5.12e3·25-s − 96.4i·27-s − 1.57e4i·29-s + 2.61e4i·31-s + ⋯
L(s)  = 1  − 1.41i·3-s + 0.819·5-s + 1.49·7-s − 0.996·9-s + 1.81·11-s + 1.71i·13-s − 1.15i·15-s + 0.318·17-s + (−0.398 − 0.916i)19-s − 2.11i·21-s − 0.0514·23-s − 0.327·25-s − 0.00489i·27-s − 0.643i·29-s + 0.879i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.398 + 0.916i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ 0.398 + 0.916i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.26934 - 1.48750i\)
\(L(\frac12)\) \(\approx\) \(2.26934 - 1.48750i\)
\(L(4)\) 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 \( 1 + (2.73e3 + 6.28e3i)T \)
good3 \( 1 + 38.1iT - 729T^{2} \)
5 \( 1 - 102.T + 1.56e4T^{2} \)
7 \( 1 - 512.T + 1.17e5T^{2} \)
11 \( 1 - 2.41e3T + 1.77e6T^{2} \)
13 \( 1 - 3.77e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.56e3T + 2.41e7T^{2} \)
23 \( 1 + 626.T + 1.48e8T^{2} \)
29 \( 1 + 1.57e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.61e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.94e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.19e5iT - 4.75e9T^{2} \)
43 \( 1 + 1.33e4T + 6.32e9T^{2} \)
47 \( 1 + 1.86e5T + 1.07e10T^{2} \)
53 \( 1 + 2.04e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.82e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.53e5T + 5.15e10T^{2} \)
67 \( 1 - 5.19e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.68e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.85e4T + 1.51e11T^{2} \)
79 \( 1 + 4.73e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.27e5T + 3.26e11T^{2} \)
89 \( 1 + 6.07e5iT - 4.96e11T^{2} \)
97 \( 1 - 2.82e5iT - 8.32e11T^{2} \)
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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

−13.23423979998528941774082824942, −11.80317066852144588193323930476, −11.45034277358976109611665016865, −9.453854943780647633886822945444, −8.407286572326488424995337547529, −7.03246730815842791717160387625, −6.25119215300870954724100826257, −4.49790604896557482901876417388, −1.90481307981278954736588639204, −1.38481190006585617389101392727, 1.52203592176866892054030876066, 3.63653121635304310063833698014, 4.87917508834811940726653087756, 5.94072944848448117514791268820, 8.008859832895877256123297594320, 9.178763416242989429124200326912, 10.16560652443244514593526788983, 11.00241644448962703920474202194, 12.19318625533498406139055447936, 13.90326215620337080595992074055

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