Properties

Label 2-76-19.7-c7-0-8
Degree $2$
Conductor $76$
Sign $-0.866 + 0.499i$
Analytic cond. $23.7412$
Root an. cond. $4.87250$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−18.8 − 32.5i)3-s + (−125. − 218. i)5-s + 965.·7-s + (385. − 667. i)9-s + 2.97e3·11-s + (4.04e3 − 7.00e3i)13-s + (−4.73e3 + 8.20e3i)15-s + (568. + 984. i)17-s + (−2.84e4 − 9.14e3i)19-s + (−1.81e4 − 3.14e4i)21-s + (−8.58e3 + 1.48e4i)23-s + (7.34e3 − 1.27e4i)25-s − 1.11e5·27-s + (−7.85e3 + 1.36e4i)29-s + 1.40e4·31-s + ⋯
L(s)  = 1  + (−0.402 − 0.696i)3-s + (−0.450 − 0.780i)5-s + 1.06·7-s + (0.176 − 0.305i)9-s + 0.674·11-s + (0.510 − 0.883i)13-s + (−0.362 + 0.627i)15-s + (0.0280 + 0.0485i)17-s + (−0.952 − 0.306i)19-s + (−0.427 − 0.741i)21-s + (−0.147 + 0.254i)23-s + (0.0940 − 0.162i)25-s − 1.08·27-s + (−0.0598 + 0.103i)29-s + 0.0848·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(23.7412\)
Root analytic conductor: \(4.87250\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :7/2),\ -0.866 + 0.499i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.372907 - 1.39242i\)
\(L(\frac12)\) \(\approx\) \(0.372907 - 1.39242i\)
\(L(\frac{9}{2})\) 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 \( 1 + (2.84e4 + 9.14e3i)T \)
good3 \( 1 + (18.8 + 32.5i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (125. + 218. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 - 965.T + 8.23e5T^{2} \)
11 \( 1 - 2.97e3T + 1.94e7T^{2} \)
13 \( 1 + (-4.04e3 + 7.00e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-568. - 984. i)T + (-2.05e8 + 3.55e8i)T^{2} \)
23 \( 1 + (8.58e3 - 1.48e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (7.85e3 - 1.36e4i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 - 1.40e4T + 2.75e10T^{2} \)
37 \( 1 + 5.52e5T + 9.49e10T^{2} \)
41 \( 1 + (7.48e4 + 1.29e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (-1.60e4 - 2.78e4i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (-4.68e5 + 8.12e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-2.88e5 + 5.00e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (5.70e5 + 9.87e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.40e6 - 2.43e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.94e5 + 3.37e5i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-5.23e5 - 9.06e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + (-1.51e6 - 2.63e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-2.11e6 - 3.66e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 7.44e6T + 2.71e13T^{2} \)
89 \( 1 + (-4.29e6 + 7.43e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + (-5.22e6 - 9.05e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−12.43574522132605331007919738431, −11.78341767838270408671398344113, −10.62153007375633588450049796107, −8.887702390728003581259056855581, −8.015055576187408699743443923487, −6.71382829990246738588401951706, −5.31280084178304524206172772773, −3.96389174961082236990002551153, −1.62562291220833284582129282941, −0.54265577092672549044799011206, 1.77213032276138049892730929376, 3.81246330084680483479841383851, 4.82458823340403366974353398327, 6.43617528225095933230950843501, 7.73821844256831330098020824190, 9.044711510559557856015230331437, 10.56461271632152454757664293561, 11.10578549659315372327656074131, 12.07644252587503946465378888671, 13.80055932704168781586968386355

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