L-function 1-837-837.295-r0-0-0

• Label: 1-837-837.295-r0-0-0
• Degree: $1$
• Conductor: $837$
• Sign: $-0.999 + 0.0433i$
• Analytic cond.: $3.88701$
• Root an. cond.: $3.88701$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.615 + 0.788i)2^-s + (−0.241 − 0.970i)4^-s + (0.173 − 0.984i)5^-s + (−0.997 + 0.0697i)7^-s + (0.913 + 0.406i)8^-s + (0.669 + 0.743i)10^-s + (0.438 − 0.898i)11^-s + (−0.615 − 0.788i)13^-s + (0.559 − 0.829i)14^-s + (−0.882 + 0.469i)16^-s + (−0.104 − 0.994i)17^-s + (−0.978 + 0.207i)19^-s + (−0.997 + 0.0697i)20^-s + (0.438 + 0.898i)22^-s + (−0.997 − 0.0697i)23^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0433i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$837$    =    $3^{3} \cdot 31$
Sign:	$-0.999 + 0.0433i$
Analytic conductor:	$3.88701$
Root analytic conductor:	$3.88701$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{837} (295, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 837,\ (0:\ ),\ -0.999 + 0.0433i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.002149267152 - 0.09912325492i$
$L(1)$	$\approx$	$0.5512557060 + 0.006408941986i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	31	$1$
good	2	$1 + (-0.615 + 0.788i)T$
	5	$1 + (0.173 - 0.984i)T$
	7	$1 + (-0.997 + 0.0697i)T$
	11	$1 + (0.438 - 0.898i)T$
	13	$1 + (-0.615 - 0.788i)T$
	17	$1 + (-0.104 - 0.994i)T$
	19	$1 + (-0.978 + 0.207i)T$
	23	$1 + (-0.997 - 0.0697i)T$
	29	$1 + (-0.615 + 0.788i)T$

Imaginary part of the first few zeros on the critical line

−22.321205025628902834265529575706, −21.82286475826558304336721294907, −20.94879604481942451142238822554, −19.89659732881566565857459920624, −19.183208299014394584516948169490, −18.95990021776518359304224716351, −17.649712670051257683444192409589, −17.3198158960734462826676672469, −16.32076070429382119086993399350, −15.31178152588499932311315932209, −14.417861764861473562021367655, −13.484160824326412078441151112983, −12.56660924243454542427475237661, −11.95860699195116351770458644036, −10.90206613503407193419001362178, −10.17515900773157530168080452780, −9.606625440277847111357303676379, −8.75471171667060159244469657487, −7.45735471703523044646956520869, −6.88929199645465093533721578297, −5.97595226833309315417900652551, −4.17575112564332487803925500297, −3.75218057865968363638669981778, −2.38192440050590045104393181592, −1.95916631330477322173917189391, 0.05848197325559463353526298396, 1.15017331864253056588113257898, 2.561234118806957877482060510577, 3.94980300468822718974308260989, 5.05403377629247561489638265205, 5.85327885020251846600656602470, 6.56082320115568375487982623308, 7.67757601093843655596546843957, 8.50524478821704606830681905223, 9.26360111807277203095022803253, 9.867071208415017598524620494673, 10.82671745759595570925561805836, 12.05461250922183930667101174292, 12.93115379989537481925332602322, 13.69075743915333565578522338032, 14.55357586299177222526334763471, 15.651931570343542213764023084437, 16.20352093158815845121251686149, 16.82922700764250676421585442493, 17.5184881749431799469762422792, 18.51846306346126055232290494375, 19.3199126920475815557985526177, 19.94150444042142454148099007446, 20.68436880426527148612789133685, 22.02848668172529673865761604673

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