# Properties

 Label 1-37-37.17-r1-0-0 Degree $1$ Conductor $37$ Sign $-0.660 + 0.750i$ Analytic cond. $3.97620$ Root an. cond. $3.97620$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.342 + 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s + i·6-s + (0.173 + 0.984i)7-s + (−0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)12-s + (0.642 + 0.766i)13-s + (−0.866 + 0.5i)14-s + (−0.984 − 0.173i)15-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + ⋯
 L(s)  = 1 + (0.342 + 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s + i·6-s + (0.173 + 0.984i)7-s + (−0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)12-s + (0.642 + 0.766i)13-s + (−0.866 + 0.5i)14-s + (−0.984 − 0.173i)15-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$37$$ Sign: $-0.660 + 0.750i$ Analytic conductor: $$3.97620$$ Root analytic conductor: $$3.97620$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{37} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 37,\ (1:\ ),\ -0.660 + 0.750i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7564683355 + 1.673815789i$$ $$L(\frac12)$$ $$\approx$$ $$0.7564683355 + 1.673815789i$$ $$L(1)$$ $$\approx$$ $$1.015147515 + 0.9818307121i$$ $$L(1)$$ $$\approx$$ $$1.015147515 + 0.9818307121i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad37 $$1$$
good2 $$1 + (0.342 + 0.939i)T$$
3 $$1 + (0.939 + 0.342i)T$$
5 $$1 + (-0.984 + 0.173i)T$$
7 $$1 + (0.173 + 0.984i)T$$
11 $$1 + (0.5 - 0.866i)T$$
13 $$1 + (0.642 + 0.766i)T$$
17 $$1 + (-0.642 + 0.766i)T$$
19 $$1 + (0.342 - 0.939i)T$$
23 $$1 + (0.866 - 0.5i)T$$
29 $$1 + (0.866 + 0.5i)T$$
31 $$1 - iT$$
41 $$1 + (-0.766 + 0.642i)T$$
43 $$1 + iT$$
47 $$1 + (-0.5 - 0.866i)T$$
53 $$1 + (0.173 - 0.984i)T$$
59 $$1 + (0.984 + 0.173i)T$$
61 $$1 + (-0.642 - 0.766i)T$$
67 $$1 + (-0.173 - 0.984i)T$$
71 $$1 + (-0.939 - 0.342i)T$$
73 $$1 - T$$
79 $$1 + (-0.984 + 0.173i)T$$
83 $$1 + (0.766 + 0.642i)T$$
89 $$1 + (-0.984 - 0.173i)T$$
97 $$1 + (0.866 - 0.5i)T$$
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$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−35.61160908983845890411086493097, −33.1928308924250546297432037432, −32.18414728337101404069295175839, −30.909747524176755451025095053159, −30.49591145131593388868497436490, −29.16981271146755239471024702601, −27.47029753945624842981810421805, −26.83571365136709561290721753873, −25.03583355221465991216180294339, −23.58157860418742240724860017578, −22.777094710871939867269641334796, −20.693008254704323044457905034519, −20.18864339286686158872813949208, −19.19193669191332450071628461892, −17.79016324212032976681197919614, −15.52418267434890532479901858654, −14.26518007539070990492994889468, −13.09632692464006284739426796567, −11.855970235655716168020647997779, −10.2834510335644586059632850546, −8.71351546369410715614686632097, −7.25962849762197406814211802986, −4.43619135004768193727228844999, −3.30227895235670864254788167464, −1.16522894113028381975445888225, 3.21010262275403156185609803370, 4.57524480880217802038040952170, 6.63584535568942885536066099347, 8.32229269293700645031532367367, 8.961796068967260042001443202769, 11.4610146427074713986373146446, 13.16969929536678622285374273570, 14.57072171488544317729391461161, 15.428134084486678071854152526811, 16.39765654454672214920403784722, 18.45256202717593330557597103210, 19.473473443476041660440451931393, 21.30059246277823398291808515113, 22.22117626150558206577433198575, 23.89462698731843285570319576492, 24.74298342104429304967160817821, 26.07971372725131298767268919292, 26.919195227815399235557475391066, 28.0463478137535074194493113921, 30.57958287681499202423529579090, 31.1380625375602056802836574429, 32.12701887831331212666082874377, 33.184586240440370836792858764396, 34.71997973360964184245714604272, 35.359502632359574692593678789847