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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 $0$

Origins

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Normalization:  

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

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