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Properties

Label 2-37-37.6-c6-0-0
Degree $2$
Conductor $37$
Sign $-0.765 - 0.642i$
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  + (−10.4 + 10.4i)2-s − 29.1i·3-s − 152. i·4-s + (−98.7 − 98.7i)5-s + (303. + 303. i)6-s − 196.·7-s + (925. + 925. i)8-s − 120.·9-s + 2.05e3·10-s + 1.70e3i·11-s − 4.45e3·12-s + (−1.61e3 − 1.61e3i)13-s + (2.04e3 − 2.04e3i)14-s + (−2.87e3 + 2.87e3i)15-s − 9.48e3·16-s + (−2.27e3 − 2.27e3i)17-s + ⋯
L(s)  = 1  + (−1.30 + 1.30i)2-s − 1.07i·3-s − 2.38i·4-s + (−0.790 − 0.790i)5-s + (1.40 + 1.40i)6-s − 0.573·7-s + (1.80 + 1.80i)8-s − 0.164·9-s + 2.05·10-s + 1.28i·11-s − 2.57·12-s + (−0.735 − 0.735i)13-s + (0.745 − 0.745i)14-s + (−0.852 + 0.852i)15-s − 2.31·16-s + (−0.462 − 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.765 - 0.642i$
Analytic conductor: \(8.51200\)
Root analytic conductor: \(2.91753\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3),\ -0.765 - 0.642i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0681519 + 0.187208i\)
\(L(\frac12)\) \(\approx\) \(0.0681519 + 0.187208i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-8.55e3 - 4.99e4i)T \)
good2 \( 1 + (10.4 - 10.4i)T - 64iT^{2} \)
3 \( 1 + 29.1iT - 729T^{2} \)
5 \( 1 + (98.7 + 98.7i)T + 1.56e4iT^{2} \)
7 \( 1 + 196.T + 1.17e5T^{2} \)
11 \( 1 - 1.70e3iT - 1.77e6T^{2} \)
13 \( 1 + (1.61e3 + 1.61e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (2.27e3 + 2.27e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (-4.72e3 - 4.72e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (-1.65e4 - 1.65e4i)T + 1.48e8iT^{2} \)
29 \( 1 + (2.24e4 - 2.24e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (2.27e4 - 2.27e4i)T - 8.87e8iT^{2} \)
41 \( 1 + 2.50e4iT - 4.75e9T^{2} \)
43 \( 1 + (6.13e4 + 6.13e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 9.06e4T + 1.07e10T^{2} \)
53 \( 1 - 2.42e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.50e5 - 1.50e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (2.01e5 - 2.01e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 1.51e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.47e5T + 1.28e11T^{2} \)
73 \( 1 - 1.21e5iT - 1.51e11T^{2} \)
79 \( 1 + (3.43e5 + 3.43e5i)T + 2.43e11iT^{2} \)
83 \( 1 + 7.95e5T + 3.26e11T^{2} \)
89 \( 1 + (1.07e5 - 1.07e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-9.80e4 - 9.80e4i)T + 8.32e11iT^{2} \)
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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

−15.76391461693106270325782601981, −14.94056359348324020605619121708, −13.18235378353482841289897797898, −12.08352052399648579733266072195, −10.03671671297749879645488044563, −8.886344432695583545728278163910, −7.43354924840184771731934335044, −7.17193459615336074382166803600, −5.25387854466366661360978846835, −1.27232372385629745574927001397, 0.17081644007410859637186381825, 2.87759969414954009069616552298, 3.98270701537025459606755448568, 7.21287158911125466332229389260, 8.823173766280116570046115229999, 9.737948597356884017060029682261, 10.95110198776582776176933586395, 11.37625489646187526699116530131, 12.98682733332481645109232195132, 14.95408778062782180490281838749

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