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Properties

Label 1-37-37.10-r0-0-0
Degree $1$
Conductor $37$
Sign $0.367 + 0.929i$
Analytic cond. $0.171827$
Root an. cond. $0.171827$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.367 + 0.929i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ 0.367 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3584808734 + 0.2437529390i\)
\(L(\frac12)\) \(\approx\) \(0.3584808734 + 0.2437529390i\)
\(L(1)\) \(\approx\) \(0.5658828146 + 0.1332196366i\)
\(L(1)\) \(\approx\) \(0.5658828146 + 0.1332196366i\)

Euler product

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

−35.35630197238204575738033224895, −34.62996326837025137546360989913, −33.00638086466986787813591827008, −32.22159200120434696928206673503, −30.53905702198002778527416820472, −29.14334227867178927027345543591, −28.05998625048205698274289856792, −27.00367140382453271971869624621, −25.3919408072716572579131014674, −24.414155312342707877482977350663, −23.50410574829866772134006533365, −22.50696190231339793682288917511, −19.82887397124643936202778866675, −19.33238163846807106143348561711, −17.32567176984484140018556660850, −17.025611398833887160251935531835, −15.47745879895944035537176708170, −13.687433814179562180921232658128, −12.53525768972267832446445680335, −10.70862050744391588524495494203, −8.855023064136236819525649104768, −7.51735135540266147294645783008, −6.354076381417678280509132185242, −4.63028954153785337382556771372, −0.898061347851247434243686946138, 2.88826231694706523648104712684, 4.347783544387761743932312815473, 6.6504143245022051931868999056, 8.88121024023511924995053081516, 9.97045583568617280977688036995, 11.391323818915167440556153289776, 12.1186496237701283764555831058, 14.41741291344390363506519278928, 15.8904641742089916390451111822, 17.191846056634922626140188840009, 18.63218015358600980809938038826, 19.6136037979251160395114244074, 21.2422565543277255693784009584, 22.20512831692100323333857471367, 22.931573605801048673347858284193, 25.343398222903553237212628429479, 26.73984316489130294318783345489, 27.332411869806509536280828616135, 28.59715907154168456447260591950, 29.54261023786679629121294729849, 31.05865764689126347441924063012, 31.96062023447427349718702192338, 33.721732297479682558937530497202, 34.78068785289530250965183119712, 35.67819108990770897690093072021

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