Properties

Label 1-37-37.21-r0-0-0
Degree $1$
Conductor $37$
Sign $0.507 - 0.861i$
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  + (−0.766 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ 0.507 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6240296764 - 0.3568470876i\)
\(L(\frac12)\) \(\approx\) \(0.6240296764 - 0.3568470876i\)
\(L(1)\) \(\approx\) \(0.8103227626 - 0.3343407036i\)
\(L(1)\) \(\approx\) \(0.8103227626 - 0.3343407036i\)

Euler product

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

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

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