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Properties

Label 1-37-37.32-r1-0-0
Degree $1$
Conductor $37$
Sign $-0.695 + 0.718i$
Analytic cond. $3.97620$
Root an. cond. $3.97620$
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  + (0.642 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ -0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4973328100 + 1.173038405i\)
\(L(\frac12)\) \(\approx\) \(0.4973328100 + 1.173038405i\)
\(L(1)\) \(\approx\) \(0.8573369617 + 0.6448789408i\)
\(L(1)\) \(\approx\) \(0.8573369617 + 0.6448789408i\)

Euler product

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

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

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