Properties

Label 2-37-37.3-c3-0-4
Degree $2$
Conductor $37$
Sign $0.979 + 0.203i$
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  + (−3.52 + 4.19i)2-s + (5.42 − 4.55i)3-s + (−3.82 − 21.7i)4-s + (4.86 − 13.3i)5-s + 38.8i·6-s + (−13.3 − 4.86i)7-s + (66.6 + 38.4i)8-s + (4.02 − 22.8i)9-s + (39.0 + 67.5i)10-s + (2.49 − 4.32i)11-s + (−119. − 100. i)12-s + (−12.4 + 2.18i)13-s + (67.5 − 38.9i)14-s + (−34.4 − 94.7i)15-s + (−230. + 83.9i)16-s + (96.2 + 16.9i)17-s + ⋯
L(s)  = 1  + (−1.24 + 1.48i)2-s + (1.04 − 0.876i)3-s + (−0.478 − 2.71i)4-s + (0.435 − 1.19i)5-s + 2.64i·6-s + (−0.721 − 0.262i)7-s + (2.94 + 1.70i)8-s + (0.149 − 0.845i)9-s + (1.23 + 2.13i)10-s + (0.0683 − 0.118i)11-s + (−2.87 − 2.41i)12-s + (−0.264 + 0.0466i)13-s + (1.28 − 0.744i)14-s + (−0.593 − 1.63i)15-s + (−3.60 + 1.31i)16-s + (1.37 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.979 + 0.203i$
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: \((2,\ 37,\ (\ :3/2),\ 0.979 + 0.203i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.911484 - 0.0935446i\)
\(L(\frac12)\) \(\approx\) \(0.911484 - 0.0935446i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (112. + 194. i)T \)
good2 \( 1 + (3.52 - 4.19i)T + (-1.38 - 7.87i)T^{2} \)
3 \( 1 + (-5.42 + 4.55i)T + (4.68 - 26.5i)T^{2} \)
5 \( 1 + (-4.86 + 13.3i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (13.3 + 4.86i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-2.49 + 4.32i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (12.4 - 2.18i)T + (2.06e3 - 751. i)T^{2} \)
17 \( 1 + (-96.2 - 16.9i)T + (4.61e3 + 1.68e3i)T^{2} \)
19 \( 1 + (-28.6 - 34.1i)T + (-1.19e3 + 6.75e3i)T^{2} \)
23 \( 1 + (-46.0 + 26.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (85.0 + 49.1i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 151. iT - 2.97e4T^{2} \)
41 \( 1 + (-66.6 - 377. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + 141. iT - 7.95e4T^{2} \)
47 \( 1 + (-105. - 183. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-423. + 154. i)T + (1.14e5 - 9.56e4i)T^{2} \)
59 \( 1 + (-102. - 282. i)T + (-1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (-688. + 121. i)T + (2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (885. + 322. i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (337. - 283. i)T + (6.21e4 - 3.52e5i)T^{2} \)
73 \( 1 + 118.T + 3.89e5T^{2} \)
79 \( 1 + (331. - 910. i)T + (-3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (124. - 706. i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (-146. - 403. i)T + (-5.40e5 + 4.53e5i)T^{2} \)
97 \( 1 + (-767. + 443. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−16.29374128370690213444188889001, −14.75709645977569134302961641773, −13.84783774656444533915460217147, −12.77886560047862663803376237187, −10.00115131336077717750947181539, −9.054404017062951507855505882443, −8.143566092170976022934474830010, −7.07787695206777322621453437189, −5.54405825179016932232967549410, −1.21714118203406256523286251618, 2.65433116051449919069216325553, 3.50069834420907430325926130898, 7.41539564894063599074622918366, 8.977701861541258653756779349356, 9.829158609891864037499163040120, 10.44934996970277461463314544315, 11.88504736017028740170908341683, 13.39327236140215118183714954623, 14.69974945774825264389910804451, 16.14026437103403376299538693329

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