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Properties

Label 2-37-37.31-c6-0-0
Degree $2$
Conductor $37$
Sign $0.770 - 0.637i$
Analytic cond. $8.51200$
Root an. cond. $2.91753$
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  + (−7.66 − 7.66i)2-s − 49.4i·3-s + 53.6i·4-s + (−147. + 147. i)5-s + (−379. + 379. i)6-s + 416.·7-s + (−79.7 + 79.7i)8-s − 1.71e3·9-s + 2.25e3·10-s − 1.71e3i·11-s + 2.64e3·12-s + (−480. + 480. i)13-s + (−3.19e3 − 3.19e3i)14-s + (7.27e3 + 7.27e3i)15-s + 4.65e3·16-s + (−2.21e3 + 2.21e3i)17-s + ⋯
L(s)  = 1  + (−0.958 − 0.958i)2-s − 1.83i·3-s + 0.837i·4-s + (−1.17 + 1.17i)5-s + (−1.75 + 1.75i)6-s + 1.21·7-s + (−0.155 + 0.155i)8-s − 2.35·9-s + 2.25·10-s − 1.29i·11-s + 1.53·12-s + (−0.218 + 0.218i)13-s + (−1.16 − 1.16i)14-s + (2.15 + 2.15i)15-s + 1.13·16-s + (−0.450 + 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(8.51200\)
Root analytic conductor: \(2.91753\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0326295 + 0.0117436i\)
\(L(\frac12)\) \(\approx\) \(0.0326295 + 0.0117436i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (8.91e3 - 4.98e4i)T \)
good2 \( 1 + (7.66 + 7.66i)T + 64iT^{2} \)
3 \( 1 + 49.4iT - 729T^{2} \)
5 \( 1 + (147. - 147. i)T - 1.56e4iT^{2} \)
7 \( 1 - 416.T + 1.17e5T^{2} \)
11 \( 1 + 1.71e3iT - 1.77e6T^{2} \)
13 \( 1 + (480. - 480. i)T - 4.82e6iT^{2} \)
17 \( 1 + (2.21e3 - 2.21e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (3.95e3 - 3.95e3i)T - 4.70e7iT^{2} \)
23 \( 1 + (90.2 - 90.2i)T - 1.48e8iT^{2} \)
29 \( 1 + (3.86e3 + 3.86e3i)T + 5.94e8iT^{2} \)
31 \( 1 + (-1.50e4 - 1.50e4i)T + 8.87e8iT^{2} \)
41 \( 1 - 9.49e4iT - 4.75e9T^{2} \)
43 \( 1 + (-2.76e4 + 2.76e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 8.07e4T + 1.07e10T^{2} \)
53 \( 1 + 1.56e5T + 2.21e10T^{2} \)
59 \( 1 + (1.43e5 - 1.43e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (2.91e5 + 2.91e5i)T + 5.15e10iT^{2} \)
67 \( 1 + 2.24e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.47e5T + 1.28e11T^{2} \)
73 \( 1 - 5.26e5iT - 1.51e11T^{2} \)
79 \( 1 + (-1.04e5 + 1.04e5i)T - 2.43e11iT^{2} \)
83 \( 1 + 2.06e5T + 3.26e11T^{2} \)
89 \( 1 + (1.39e5 + 1.39e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (8.22e5 - 8.22e5i)T - 8.32e11iT^{2} \)
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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

−14.87438483114351507983902327152, −13.97336405910006361385589561943, −12.26726523093298951552055983149, −11.36242334494057372188467503460, −10.95177893903060926564833620809, −8.323593992345460601345010687206, −7.941804451187823009971920747581, −6.41003094070383175726228080705, −2.95591473553117958359313580450, −1.50826626274923323518603073705, 0.02456540970752535389408511353, 4.25273052592908065815374590980, 5.04255843801731036927419553137, 7.67290697221625635172851618218, 8.666344699395524009186254056947, 9.472861326306541239056657453873, 10.94235231678036728711646475204, 12.20466849421929194973594485981, 14.76217700301551511949523182215, 15.39895373235954535253436740263

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