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Properties

Label 2-76-19.11-c5-0-6
Degree $2$
Conductor $76$
Sign $-0.444 + 0.895i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.99 + 6.92i)3-s + (31.6 − 54.7i)5-s − 80.2·7-s + (89.5 + 155. i)9-s − 475.·11-s + (−337. − 584. i)13-s + (252. + 437. i)15-s + (866. − 1.50e3i)17-s + (−574. − 1.46e3i)19-s + (320. − 555. i)21-s + (−2.42e3 − 4.19e3i)23-s + (−435. − 754. i)25-s − 3.37e3·27-s + (2.39e3 + 4.14e3i)29-s − 127.·31-s + ⋯
L(s)  = 1  + (−0.256 + 0.444i)3-s + (0.565 − 0.979i)5-s − 0.619·7-s + (0.368 + 0.638i)9-s − 1.18·11-s + (−0.553 − 0.959i)13-s + (0.289 + 0.502i)15-s + (0.727 − 1.25i)17-s + (−0.365 − 0.931i)19-s + (0.158 − 0.274i)21-s + (−0.955 − 1.65i)23-s + (−0.139 − 0.241i)25-s − 0.890·27-s + (0.528 + 0.915i)29-s − 0.0237·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.475473 - 0.766803i\)
\(L(\frac12)\) \(\approx\) \(0.475473 - 0.766803i\)
\(L(\frac{7}{2})\) 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 \( 1 + (574. + 1.46e3i)T \)
good3 \( 1 + (3.99 - 6.92i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-31.6 + 54.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + 80.2T + 1.68e4T^{2} \)
11 \( 1 + 475.T + 1.61e5T^{2} \)
13 \( 1 + (337. + 584. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-866. + 1.50e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (2.42e3 + 4.19e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-2.39e3 - 4.14e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + 127.T + 2.86e7T^{2} \)
37 \( 1 + 1.39e4T + 6.93e7T^{2} \)
41 \( 1 + (-7.88e3 + 1.36e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-964. + 1.66e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-8.09e3 - 1.40e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-8.92e3 - 1.54e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-8.42e3 + 1.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.16e4 - 2.01e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.36e4 - 2.35e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (3.74e4 - 6.48e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-3.49e4 + 6.04e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.14e4 + 1.97e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 5.80e4T + 3.93e9T^{2} \)
89 \( 1 + (-8.20e3 - 1.42e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (-6.57e4 + 1.13e5i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−13.00168846776232759791712491829, −12.40727934430008039047417585334, −10.60117411668347855215711736842, −9.975518058258335306780871167258, −8.715069833574589341129686141051, −7.34674789458119622283057084561, −5.49813861317164991140069900097, −4.78736784901052460730422415594, −2.61051429377382380956808883238, −0.38098244169035234654717113735, 1.95312470781563132457325502289, 3.61677734259114824910701134728, 5.80760562098712252536590403248, 6.67323480862749638844326775689, 7.87196192779216149802608768646, 9.757772244124497397352347122002, 10.31757008991812066363081695411, 11.82933018798837661020726192004, 12.77860098417974201431302458185, 13.83012067008965083522939166038

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