Properties

Degree 1
Conductor 37
Sign $-0.695 - 0.718i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.695 - 0.718i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.695 - 0.718i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(37\)
\( \varepsilon \)  =  $-0.695 - 0.718i$
motivic weight  =  \(0\)
character  :  $\chi_{37} (22, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 37,\ (1:\ ),\ -0.695 - 0.718i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4973328100 - 1.173038405i$
$L(\frac12,\chi)$  $\approx$  $0.4973328100 - 1.173038405i$
$L(\chi,1)$  $\approx$  0.8573369617 - 0.6448789408i
$L(1,\chi)$  $\approx$  0.8573369617 - 0.6448789408i

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

−35.27567303960655038813063733894, −34.52014430558602859627411824522, −33.56532043694587254637654932162, −32.39817642860280352731872162108, −30.85712468053556576457950532155, −29.95864681081444899778149673032, −28.88575782745006127693121028072, −27.10392961609987645734471979995, −25.592439426116743568273865987699, −24.916860315014906911277298127345, −23.28766660084352789096772594389, −22.54928393844034253463090994356, −21.72187031978059630662576465051, −19.37155133190820024105941721735, −17.97350435746563840096666908380, −17.02840057060060584305305478834, −15.549513996519827405703254116514, −14.1847325803349852656838067085, −12.82187775305803092446864519735, −11.77418782512903849358461507184, −9.77879515577809560033134352666, −7.34532620896352088547247513986, −6.57869448505227478144465551739, −5.201351003068377211836075124923, −2.8637688672396449529425135809, 0.77129842537470962456694517118, 3.52594744220396523294098996491, 5.01338752816848704794670409359, 6.21324931433848157046600453621, 9.32279751145666447259101020047, 10.21943722883182870260017919192, 11.84916127356179936583675045897, 12.79961850261177182518224138073, 14.336559468531990111026907303, 16.11475430587430754864588252694, 17.03266930304126799980671161746, 18.99620532982969522029345306175, 20.32617808402004149660913144403, 21.43790755882075503135163159915, 22.39239898393087000478635170767, 23.529173612981751669936295253196, 24.76766092722447209458336311291, 26.85266772650956766226493365451, 27.94084189415491466950432792180, 29.18065491317563596544848882138, 29.49010383523969917599856987575, 31.645710270851703304861242454988, 32.4754477695787857478531935404, 33.141130796720894681784640560, 34.779768897238362195745401152393

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