Properties

Label 2-76-19.17-c5-0-2
Degree $2$
Conductor $76$
Sign $-0.626 - 0.779i$
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  + (22.5 + 18.9i)3-s + (−48.8 + 17.7i)5-s + (−11.8 − 20.5i)7-s + (107. + 612. i)9-s + (−249. + 431. i)11-s + (58.3 − 48.9i)13-s + (−1.43e3 − 523. i)15-s + (282. − 1.59e3i)17-s + (−8.67 + 1.57e3i)19-s + (120. − 685. i)21-s + (215. + 78.2i)23-s + (−320. + 268. i)25-s + (−5.56e3 + 9.64e3i)27-s + (−663. − 3.76e3i)29-s + (3.29e3 + 5.71e3i)31-s + ⋯
L(s)  = 1  + (1.44 + 1.21i)3-s + (−0.874 + 0.318i)5-s + (−0.0913 − 0.158i)7-s + (0.444 + 2.51i)9-s + (−0.620 + 1.07i)11-s + (0.0957 − 0.0803i)13-s + (−1.64 − 0.600i)15-s + (0.236 − 1.34i)17-s + (−0.00551 + 0.999i)19-s + (0.0598 − 0.339i)21-s + (0.0847 + 0.0308i)23-s + (−0.102 + 0.0860i)25-s + (−1.46 + 2.54i)27-s + (−0.146 − 0.830i)29-s + (0.616 + 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.626 - 0.779i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ -0.626 - 0.779i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.917848 + 1.91477i\)
\(L(\frac12)\) \(\approx\) \(0.917848 + 1.91477i\)
\(L(\frac{7}{2})\) 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 \( 1 + (8.67 - 1.57e3i)T \)
good3 \( 1 + (-22.5 - 18.9i)T + (42.1 + 239. i)T^{2} \)
5 \( 1 + (48.8 - 17.7i)T + (2.39e3 - 2.00e3i)T^{2} \)
7 \( 1 + (11.8 + 20.5i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (249. - 431. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-58.3 + 48.9i)T + (6.44e4 - 3.65e5i)T^{2} \)
17 \( 1 + (-282. + 1.59e3i)T + (-1.33e6 - 4.85e5i)T^{2} \)
23 \( 1 + (-215. - 78.2i)T + (4.93e6 + 4.13e6i)T^{2} \)
29 \( 1 + (663. + 3.76e3i)T + (-1.92e7 + 7.01e6i)T^{2} \)
31 \( 1 + (-3.29e3 - 5.71e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.15e4T + 6.93e7T^{2} \)
41 \( 1 + (-1.51e4 - 1.27e4i)T + (2.01e7 + 1.14e8i)T^{2} \)
43 \( 1 + (3.67e3 - 1.33e3i)T + (1.12e8 - 9.44e7i)T^{2} \)
47 \( 1 + (2.75e3 + 1.56e4i)T + (-2.15e8 + 7.84e7i)T^{2} \)
53 \( 1 + (1.67e4 + 6.09e3i)T + (3.20e8 + 2.68e8i)T^{2} \)
59 \( 1 + (71.6 - 406. i)T + (-6.71e8 - 2.44e8i)T^{2} \)
61 \( 1 + (-4.09e4 - 1.48e4i)T + (6.46e8 + 5.42e8i)T^{2} \)
67 \( 1 + (-5.36e3 - 3.04e4i)T + (-1.26e9 + 4.61e8i)T^{2} \)
71 \( 1 + (-5.04e4 + 1.83e4i)T + (1.38e9 - 1.15e9i)T^{2} \)
73 \( 1 + (1.99e4 + 1.67e4i)T + (3.59e8 + 2.04e9i)T^{2} \)
79 \( 1 + (-2.96e4 - 2.48e4i)T + (5.34e8 + 3.03e9i)T^{2} \)
83 \( 1 + (6.30e3 + 1.09e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-6.29e4 + 5.27e4i)T + (9.69e8 - 5.49e9i)T^{2} \)
97 \( 1 + (-1.56e4 + 8.85e4i)T + (-8.06e9 - 2.93e9i)T^{2} \)
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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

−14.25001785484912488989747310387, −13.09784251936863924924097790490, −11.53928269300586497330216152311, −10.21240352517359261158590008256, −9.561780180373268129886921150442, −8.152263401705652848790051142431, −7.43665720990392869614616248408, −4.86000865111437630660410621831, −3.77843168703841229953636068507, −2.58709161992189134149339910944, 0.792381745356484340994220619705, 2.58081772598901249548502013949, 3.85007125137493106016646233503, 6.23141251321461398837251041337, 7.69125194819306708124281660812, 8.263248438619972353495132841544, 9.231819548108704245480877471801, 11.13447589566434307788797549944, 12.46155183136264280566930966482, 13.10508287480884836348292483167

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