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Properties

Label 2-37-37.6-c6-0-9
Degree $2$
Conductor $37$
Sign $0.629 + 0.777i$
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  + (−8.48 + 8.48i)2-s + 25.9i·3-s − 80.1i·4-s + (25.7 + 25.7i)5-s + (−219. − 219. i)6-s − 623.·7-s + (136. + 136. i)8-s + 57.8·9-s − 436.·10-s − 647. i·11-s + 2.07e3·12-s + (691. + 691. i)13-s + (5.29e3 − 5.29e3i)14-s + (−666. + 666. i)15-s + 2.80e3·16-s + (−621. − 621. i)17-s + ⋯
L(s)  = 1  + (−1.06 + 1.06i)2-s + 0.959i·3-s − 1.25i·4-s + (0.205 + 0.205i)5-s + (−1.01 − 1.01i)6-s − 1.81·7-s + (0.267 + 0.267i)8-s + 0.0793·9-s − 0.436·10-s − 0.486i·11-s + 1.20·12-s + (0.314 + 0.314i)13-s + (1.92 − 1.92i)14-s + (−0.197 + 0.197i)15-s + 0.685·16-s + (−0.126 − 0.126i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.629 + 0.777i$
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: \((2,\ 37,\ (\ :3),\ 0.629 + 0.777i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0408674 - 0.0195010i\)
\(L(\frac12)\) \(\approx\) \(0.0408674 - 0.0195010i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.13e3 + 5.06e4i)T \)
good2 \( 1 + (8.48 - 8.48i)T - 64iT^{2} \)
3 \( 1 - 25.9iT - 729T^{2} \)
5 \( 1 + (-25.7 - 25.7i)T + 1.56e4iT^{2} \)
7 \( 1 + 623.T + 1.17e5T^{2} \)
11 \( 1 + 647. iT - 1.77e6T^{2} \)
13 \( 1 + (-691. - 691. i)T + 4.82e6iT^{2} \)
17 \( 1 + (621. + 621. i)T + 2.41e7iT^{2} \)
19 \( 1 + (-3.14e3 - 3.14e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (1.01e4 + 1.01e4i)T + 1.48e8iT^{2} \)
29 \( 1 + (9.14e3 - 9.14e3i)T - 5.94e8iT^{2} \)
31 \( 1 + (9.01e3 - 9.01e3i)T - 8.87e8iT^{2} \)
41 \( 1 + 5.79e4iT - 4.75e9T^{2} \)
43 \( 1 + (9.83e4 + 9.83e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 1.70e4T + 1.07e10T^{2} \)
53 \( 1 + 2.39e5T + 2.21e10T^{2} \)
59 \( 1 + (8.97e4 + 8.97e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (5.48e4 - 5.48e4i)T - 5.15e10iT^{2} \)
67 \( 1 - 1.94e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.20e5T + 1.28e11T^{2} \)
73 \( 1 - 4.40e5iT - 1.51e11T^{2} \)
79 \( 1 + (1.29e5 + 1.29e5i)T + 2.43e11iT^{2} \)
83 \( 1 + 5.52e5T + 3.26e11T^{2} \)
89 \( 1 + (4.12e5 - 4.12e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-1.07e6 - 1.07e6i)T + 8.32e11iT^{2} \)
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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

−15.65617601519872663176536190371, −14.19254568013787489031379522470, −12.60566213761106725202993936342, −10.42132752453376980317819289416, −9.729249750055087629863079297955, −8.772788551363483562262129121904, −6.97751316402355312068837932787, −5.93811949409337611468191122847, −3.57956097618774714604062882292, −0.03246917376336139819580158874, 1.51087028246222623529308072212, 3.15169287567915060773453340924, 6.28163922590896278779648659600, 7.71005510072059411916779700137, 9.372335251110268702797603273168, 10.00767328688527895897536914166, 11.63957921806467886939950998302, 12.78214186696684307230859903761, 13.32183524012423068409676615477, 15.55213013628798374804825319171

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