Properties

Label 1-37-37.16-r0-0-0
Degree $1$
Conductor $37$
Sign $0.918 + 0.395i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.918 + 0.395i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ 0.918 + 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123945752 + 0.2318234829i\)
\(L(\frac12)\) \(\approx\) \(1.123945752 + 0.2318234829i\)
\(L(1)\) \(\approx\) \(1.347420239 + 0.2287507207i\)
\(L(1)\) \(\approx\) \(1.347420239 + 0.2287507207i\)

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

−35.591005891246716537339542284907, −34.1260552475515047475748776447, −32.56058697222732123543582988990, −31.920344180466605436220985914107, −30.9933927822704909778004954614, −29.8344443490234432809352337284, −28.31421453516343753792432814698, −27.21396259929731184328147628795, −25.93334213200739237931276883996, −24.42760485373204856798483069660, −22.93252993072868225392483944010, −22.03461877886432147840840443950, −20.74871436036363558226284951216, −19.54182888502293959624360037605, −18.88295180225782784345203103884, −15.92157181570684824718890933585, −15.355396938052683991176612841700, −13.90094751931467527280910022890, −12.594092042769895040719455731119, −10.9811524114810998824101652566, −9.82422325404995403762387951091, −8.0235563851890551763504913354, −5.75985912152735815427950173459, −3.78797158344529489509425318575, −2.967734476261450887825855095698, 2.98263622148487153915444197170, 4.4460802596376722671198145886, 6.765866126230916827898516123808, 7.64113175537467949013766673853, 9.17730538607335376462320652680, 11.895926931418480777239949652663, 12.89178225691047354786806851804, 14.045796791726876988710876241740, 15.460526412939500344942647906039, 16.428492328543676287268762818774, 18.31220910999250330441608291573, 19.824548167775281700992456078629, 20.72885793066942896324800183383, 22.66846474768538584699689606269, 23.58878149714085156211066988945, 24.577349248285784270916030408058, 25.89515557935765105374071538156, 26.6256749590172552109719102586, 28.73575458384905455903997831915, 30.16997334636953014283628862830, 31.23062148169681445397521129665, 31.84538321753655712033758926590, 33.0696165797525618153859096930, 34.61563161729989997325206011140, 35.7703690424167952449320709890

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