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Properties

Label 2-37-37.27-c3-0-3
Degree $2$
Conductor $37$
Sign $0.827 + 0.561i$
Analytic cond. $2.18307$
Root an. cond. $1.47752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.261 − 0.150i)2-s + (0.675 + 1.17i)3-s + (−3.95 − 6.84i)4-s + (17.4 − 10.0i)5-s − 0.408i·6-s + (10.5 + 18.3i)7-s + 4.80i·8-s + (12.5 − 21.7i)9-s − 6.06·10-s − 53.3·11-s + (5.34 − 9.25i)12-s + (−9.28 + 5.35i)13-s − 6.39i·14-s + (23.5 + 13.5i)15-s + (−30.9 + 53.5i)16-s + (58.6 + 33.8i)17-s + ⋯
L(s)  = 1  + (−0.0924 − 0.0533i)2-s + (0.130 + 0.225i)3-s + (−0.494 − 0.856i)4-s + (1.55 − 0.898i)5-s − 0.0277i·6-s + (0.571 + 0.990i)7-s + 0.212i·8-s + (0.466 − 0.807i)9-s − 0.191·10-s − 1.46·11-s + (0.128 − 0.222i)12-s + (−0.197 + 0.114i)13-s − 0.122i·14-s + (0.405 + 0.233i)15-s + (−0.482 + 0.836i)16-s + (0.836 + 0.483i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33289 - 0.409432i\)
\(L(\frac12)\) \(\approx\) \(1.33289 - 0.409432i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (165. - 152. i)T \)
good2 \( 1 + (0.261 + 0.150i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-0.675 - 1.17i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-17.4 + 10.0i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-10.5 - 18.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 53.3T + 1.33e3T^{2} \)
13 \( 1 + (9.28 - 5.35i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-58.6 - 33.8i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (59.0 - 34.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 137. iT - 1.21e4T^{2} \)
29 \( 1 - 202. iT - 2.43e4T^{2} \)
31 \( 1 + 214. iT - 2.97e4T^{2} \)
41 \( 1 + (89.4 + 155. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 173. iT - 7.95e4T^{2} \)
47 \( 1 + 84.7T + 1.03e5T^{2} \)
53 \( 1 + (193. - 335. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-109. - 63.1i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-380. + 219. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (171. + 296. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-0.101 - 0.176i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 1.97T + 3.89e5T^{2} \)
79 \( 1 + (28.7 - 16.6i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-225. + 390. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (150. + 86.8i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 829. iT - 9.12e5T^{2} \)
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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

−15.57314292313160829581735946168, −14.62166334481099498680379976249, −13.41689904527093459069575357294, −12.44619402581480773484183975871, −10.37321114512868396096723079309, −9.544275171545454520179796038003, −8.560806492899937194590835690128, −5.82414756587157928071026637876, −5.10506777327499140261548402088, −1.75191940294266513557964419429, 2.55714457786963813619454722056, 4.96420418681122106530278790878, 7.02592775113355749169899499472, 8.095629986542517804636597621566, 10.00214090266433072155532814244, 10.71378528544808420668013955384, 12.89844077677297361835783392461, 13.59045952324943581507396421035, 14.39006135121548574441883296333, 16.31179489914347408666390141317

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