Properties

Degree 1
Conductor 37
Sign $0.918 - 0.395i$
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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $0.918 - 0.395i$
motivic weight  =  \(0\)
character  :  $\chi_{37} (7, \cdot )$
Sato-Tate  :  $\mu(9)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 37,\ (0:\ ),\ 0.918 - 0.395i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.123945752 - 0.2318234829i$
$L(\frac12,\chi)$  $\approx$  $1.123945752 - 0.2318234829i$
$L(\chi,1)$  $\approx$  1.347420239 - 0.2287507207i
$L(1,\chi)$  $\approx$  1.347420239 - 0.2287507207i

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

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

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