Properties

Degree 1
Conductor 37
Sign $0.507 + 0.861i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.507 + 0.861i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.507 + 0.861i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $0.507 + 0.861i$
motivic weight  =  \(0\)
character  :  $\chi_{37} (30, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 37,\ (0:\ ),\ 0.507 + 0.861i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6240296764 + 0.3568470876i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6240296764 + 0.3568470876i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8103227626 + 0.3343407036i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8103227626 + 0.3343407036i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−35.808997612544623126615346466034, −34.64308497712416403045691775548, −32.980323766025484542241413272433, −31.54919727290378599488257578951, −30.20280893463774179917116719112, −29.54952867029560153271068927924, −28.48822662709977640161006843423, −26.71520395375193252720548625163, −25.64912921021142249421599110283, −25.2144885254329495771198987726, −23.06185427006406983483514411021, −21.51383228035386598228110638864, −20.306020143682964080134646710890, −19.30968169880081102662704560327, −18.126414708898393387547717510661, −17.14206558514237649831133392174, −15.13700216056068021269484392335, −13.26514695196837191853989466575, −12.66659678608183517102203968952, −10.389727315674736710335476755022, −9.51333882437292896014725388815, −7.90312777864452880887087604839, −6.53983082325083367155059845195, −3.32454339169609036855260480600, −1.95336696118314440036871325807, 2.49646006404467100386676971347, 5.09506737703336419862193423801, 6.695267684449725010926196862028, 8.73643280299977340213003430627, 9.379903377004036466245165740536, 10.66793824595772740368608916622, 13.32005944047160282448380383911, 14.444518460482160062573268937650, 15.98991801716090136655066195867, 16.66748746945026430258019818887, 18.46143258596286091294611502032, 19.521776229607043180356025294022, 20.917547447856203934530155247502, 22.14489938471707453137930006726, 24.11062840752429180948030082937, 25.2952034443550359880874628829, 26.00618266419713461499698317649, 27.04270847772203584508678087285, 28.520311123459809806755111471883, 29.33742077454344383573773046381, 31.59594720565937586338425012834, 32.41733717884133261940449558367, 33.3525766270893292987028566054, 34.54626378388510505160643366943, 36.16980325720836027863839250819

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