Properties

Label 2-37-37.3-c3-0-0
Degree $2$
Conductor $37$
Sign $-0.611 + 0.791i$
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  + (−2.33 + 2.77i)2-s + (−5.31 + 4.45i)3-s + (−0.894 − 5.07i)4-s + (2.98 − 8.19i)5-s − 25.1i·6-s + (−5.48 − 1.99i)7-s + (−8.94 − 5.16i)8-s + (3.66 − 20.7i)9-s + (15.8 + 27.3i)10-s + (−14.6 + 25.3i)11-s + (27.3 + 22.9i)12-s + (−37.8 + 6.66i)13-s + (18.3 − 10.5i)14-s + (20.6 + 56.8i)15-s + (73.9 − 26.9i)16-s + (−116. − 20.4i)17-s + ⋯
L(s)  = 1  + (−0.824 + 0.982i)2-s + (−1.02 + 0.858i)3-s + (−0.111 − 0.634i)4-s + (0.266 − 0.733i)5-s − 1.71i·6-s + (−0.296 − 0.107i)7-s + (−0.395 − 0.228i)8-s + (0.135 − 0.769i)9-s + (0.500 + 0.866i)10-s + (−0.401 + 0.695i)11-s + (0.658 + 0.552i)12-s + (−0.806 + 0.142i)13-s + (0.349 − 0.201i)14-s + (0.356 + 0.978i)15-s + (1.15 − 0.420i)16-s + (−1.65 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.611 + 0.791i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ -0.611 + 0.791i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0972890 - 0.198034i\)
\(L(\frac12)\) \(\approx\) \(0.0972890 - 0.198034i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-224. + 15.9i)T \)
good2 \( 1 + (2.33 - 2.77i)T + (-1.38 - 7.87i)T^{2} \)
3 \( 1 + (5.31 - 4.45i)T + (4.68 - 26.5i)T^{2} \)
5 \( 1 + (-2.98 + 8.19i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (5.48 + 1.99i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (14.6 - 25.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (37.8 - 6.66i)T + (2.06e3 - 751. i)T^{2} \)
17 \( 1 + (116. + 20.4i)T + (4.61e3 + 1.68e3i)T^{2} \)
19 \( 1 + (-74.3 - 88.5i)T + (-1.19e3 + 6.75e3i)T^{2} \)
23 \( 1 + (66.1 - 38.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (134. + 77.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 62.0iT - 2.97e4T^{2} \)
41 \( 1 + (-51.0 - 289. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 - 407. iT - 7.95e4T^{2} \)
47 \( 1 + (-155. - 269. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-621. + 226. i)T + (1.14e5 - 9.56e4i)T^{2} \)
59 \( 1 + (117. + 323. i)T + (-1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (500. - 88.2i)T + (2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (419. + 152. i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (435. - 365. i)T + (6.21e4 - 3.52e5i)T^{2} \)
73 \( 1 + 317.T + 3.89e5T^{2} \)
79 \( 1 + (-240. + 660. i)T + (-3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (-100. + 571. i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (268. + 737. i)T + (-5.40e5 + 4.53e5i)T^{2} \)
97 \( 1 + (364. - 210. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−16.52941970683147089528959242944, −16.04152112912243133537704460179, −14.94624605087790672770875420878, −12.98014204451520643038875103954, −11.65961279433024389293113782755, −10.02105116209503751970947195045, −9.292371004198411377313910438333, −7.60086222797601835734675048253, −6.04392413910956484265626498875, −4.71909969426617796682228526665, 0.26255622113986631831164953668, 2.48155984893366061858054511481, 5.80934607675046304158302723180, 7.10948313006373475701527228748, 9.033433349612669540353033642287, 10.52314409601559259452579024664, 11.25558395251828401450685018257, 12.28991850647660433211284065449, 13.51236156838354100520815266747, 15.22104452308157420309751133121

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