Properties

Label 2-76-19.7-c5-0-6
Degree $2$
Conductor $76$
Sign $0.535 + 0.844i$
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  + (4.20 + 7.28i)3-s + (−18.6 − 32.3i)5-s + 15.8·7-s + (86.1 − 149. i)9-s − 325.·11-s + (519. − 900. i)13-s + (156. − 271. i)15-s + (778. + 1.34e3i)17-s + (418. − 1.51e3i)19-s + (66.7 + 115. i)21-s + (784. − 1.35e3i)23-s + (866. − 1.50e3i)25-s + 3.49e3·27-s + (4.02e3 − 6.96e3i)29-s − 9.52e3·31-s + ⋯
L(s)  = 1  + (0.269 + 0.467i)3-s + (−0.333 − 0.577i)5-s + 0.122·7-s + (0.354 − 0.614i)9-s − 0.811·11-s + (0.852 − 1.47i)13-s + (0.179 − 0.311i)15-s + (0.653 + 1.13i)17-s + (0.265 − 0.964i)19-s + (0.0330 + 0.0572i)21-s + (0.309 − 0.535i)23-s + (0.277 − 0.480i)25-s + 0.921·27-s + (0.888 − 1.53i)29-s − 1.78·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.50532 - 0.827660i\)
\(L(\frac12)\) \(\approx\) \(1.50532 - 0.827660i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-418. + 1.51e3i)T \)
good3 \( 1 + (-4.20 - 7.28i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (18.6 + 32.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 15.8T + 1.68e4T^{2} \)
11 \( 1 + 325.T + 1.61e5T^{2} \)
13 \( 1 + (-519. + 900. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-778. - 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (-784. + 1.35e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-4.02e3 + 6.96e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 9.52e3T + 2.86e7T^{2} \)
37 \( 1 + 451.T + 6.93e7T^{2} \)
41 \( 1 + (-2.23e3 - 3.86e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-5.77e3 - 1.00e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (3.08e3 - 5.33e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.11e4 - 1.93e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (3.51e3 + 6.08e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-4.07e3 + 7.05e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.73e4 - 4.73e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-2.48e4 - 4.30e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (2.91e4 + 5.05e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.83e4 + 3.17e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.56e4T + 3.93e9T^{2} \)
89 \( 1 + (2.01e4 - 3.48e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (5.67e3 + 9.83e3i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.10336614593101477450842452747, −12.52457339771181366314837779966, −10.97099227896417895261573709748, −10.02079956339093173067727231281, −8.695258851617263370394072619590, −7.81310793382519403377217206749, −5.98620960124067733786235127116, −4.55336621695638505064719382300, −3.16295556990512300923438160219, −0.77803672007676505450226983611, 1.68269382467910362883224066392, 3.37510417437139659585468789339, 5.15586200310962448801921939955, 6.92535257761383429735891021768, 7.70128610490026688101850724543, 9.066086120378708331127987251923, 10.52369033172684977483654602649, 11.45154489101804222715861617951, 12.69517747782040550537518207968, 13.82986840688796654342516488770

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