L-function 1-37-37.32-r1-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	37	$1$
good	2	$1 + (0.642 + 0.766i)T$
	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	$1 + (0.866 + 0.5i)T$

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