Properties

Label 2-76-19.7-c5-0-8
Degree $2$
Conductor $76$
Sign $-0.104 - 0.994i$
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  + (−11.6 − 20.2i)3-s + (−25.2 − 43.7i)5-s − 187.·7-s + (−150. + 261. i)9-s + 411.·11-s + (286. − 496. i)13-s + (−589. + 1.02e3i)15-s + (559. + 968. i)17-s + (−1.04e3 + 1.17e3i)19-s + (2.19e3 + 3.79e3i)21-s + (−72.0 + 124. i)23-s + (287. − 498. i)25-s + 1.36e3·27-s + (−2.61e3 + 4.53e3i)29-s − 6.72e3·31-s + ⋯
L(s)  = 1  + (−0.748 − 1.29i)3-s + (−0.451 − 0.782i)5-s − 1.44·7-s + (−0.620 + 1.07i)9-s + 1.02·11-s + (0.470 − 0.815i)13-s + (−0.676 + 1.17i)15-s + (0.469 + 0.812i)17-s + (−0.666 + 0.745i)19-s + (1.08 + 1.87i)21-s + (−0.0283 + 0.0491i)23-s + (0.0920 − 0.159i)25-s + 0.361·27-s + (−0.577 + 1.00i)29-s − 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.104 - 0.994i$
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: \((2,\ 76,\ (\ :5/2),\ -0.104 - 0.994i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0611710 + 0.0679336i\)
\(L(\frac12)\) \(\approx\) \(0.0611710 + 0.0679336i\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.04e3 - 1.17e3i)T \)
good3 \( 1 + (11.6 + 20.2i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (25.2 + 43.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + 187.T + 1.68e4T^{2} \)
11 \( 1 - 411.T + 1.61e5T^{2} \)
13 \( 1 + (-286. + 496. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-559. - 968. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (72.0 - 124. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (2.61e3 - 4.53e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 6.72e3T + 2.86e7T^{2} \)
37 \( 1 - 1.23e4T + 6.93e7T^{2} \)
41 \( 1 + (5.96e3 + 1.03e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (2.61e3 + 4.52e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.04e4 - 1.81e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-4.55e3 + 7.88e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.20e4 + 2.08e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.15e4 - 3.73e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (8.05e3 - 1.39e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (3.99e4 + 6.91e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (1.72e4 + 2.98e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.63e4 - 2.83e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 4.96e4T + 3.93e9T^{2} \)
89 \( 1 + (3.92e4 - 6.79e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-6.26e4 - 1.08e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.63044230496250639329365919285, −12.11486622068148351443532496451, −10.68819085366734927469523047345, −9.156004223925523002221610641141, −7.86884798928262814261385467253, −6.57744541421206514184382964359, −5.79017759614981075203738373493, −3.68618482059273039113482592102, −1.32904603744574112933698201738, −0.04719071788790408442699789061, 3.29063369164757814019811246684, 4.28830175329479617562085892603, 6.04383453747867773983639570374, 6.97190421568850382056334956392, 9.205019451150144903485770477148, 9.833362914503435061047290218897, 11.08699115378154787456024907152, 11.69383429832478537407894442937, 13.21973912471081576407533710494, 14.66271299918241325017366655706

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