Properties

Label 2-76-19.18-c6-0-5
Degree $2$
Conductor $76$
Sign $0.147 + 0.989i$
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  + 40.3i·3-s − 186.·5-s + 202.·7-s − 898.·9-s − 1.59e3·11-s + 143. i·13-s − 7.53e3i·15-s + 4.77e3·17-s + (−1.01e3 − 6.78e3i)19-s + 8.16e3i·21-s + 7.73e3·23-s + 1.92e4·25-s − 6.83e3i·27-s + 8.82e3i·29-s − 5.15e4i·31-s + ⋯
L(s)  = 1  + 1.49i·3-s − 1.49·5-s + 0.589·7-s − 1.23·9-s − 1.19·11-s + 0.0652i·13-s − 2.23i·15-s + 0.971·17-s + (−0.147 − 0.989i)19-s + 0.881i·21-s + 0.635·23-s + 1.23·25-s − 0.347i·27-s + 0.361i·29-s − 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.147 + 0.989i$
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: \((2,\ 76,\ (\ :3),\ 0.147 + 0.989i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.187357 - 0.161518i\)
\(L(\frac12)\) \(\approx\) \(0.187357 - 0.161518i\)
\(L(4)\) 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 \( 1 + (1.01e3 + 6.78e3i)T \)
good3 \( 1 - 40.3iT - 729T^{2} \)
5 \( 1 + 186.T + 1.56e4T^{2} \)
7 \( 1 - 202.T + 1.17e5T^{2} \)
11 \( 1 + 1.59e3T + 1.77e6T^{2} \)
13 \( 1 - 143. iT - 4.82e6T^{2} \)
17 \( 1 - 4.77e3T + 2.41e7T^{2} \)
23 \( 1 - 7.73e3T + 1.48e8T^{2} \)
29 \( 1 - 8.82e3iT - 5.94e8T^{2} \)
31 \( 1 + 5.15e4iT - 8.87e8T^{2} \)
37 \( 1 + 9.46e4iT - 2.56e9T^{2} \)
41 \( 1 - 7.87e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.29e5T + 6.32e9T^{2} \)
47 \( 1 + 9.72e4T + 1.07e10T^{2} \)
53 \( 1 + 4.61e4iT - 2.21e10T^{2} \)
59 \( 1 - 6.17e4iT - 4.21e10T^{2} \)
61 \( 1 + 8.56e4T + 5.15e10T^{2} \)
67 \( 1 + 4.90e4iT - 9.04e10T^{2} \)
71 \( 1 - 4.08e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.30e5T + 1.51e11T^{2} \)
79 \( 1 + 1.06e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.22e4T + 3.26e11T^{2} \)
89 \( 1 + 5.61e5iT - 4.96e11T^{2} \)
97 \( 1 + 8.51e5iT - 8.32e11T^{2} \)
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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

−12.96353956266022525185840016262, −11.49515246172940547217160330113, −10.96172415365668557215206806695, −9.785941567124836651944950121113, −8.445897983097858856999705937342, −7.51353674409338609420878058380, −5.22457789001735216481205751997, −4.33644624358380704148085202061, −3.14135421653420517638554082514, −0.099918861017052233589202121178, 1.37457836824521052146153700663, 3.19618052833142988101970536198, 5.08627815986691194357724860715, 6.81579948605464028420962297232, 7.961642605726728131620105117031, 8.158683876762347505190363806333, 10.50154751867358037301088884852, 11.76380754910992687093295778237, 12.30864949331904822117759011145, 13.33433679934479817290468659184

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