Properties

Degree 1
Conductor $ 2^{2} \cdot 23 $
Sign $0.178 + 0.983i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.654 + 0.755i)3-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.959 + 0.281i)25-s + (−0.841 + 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + ⋯
L(s,χ)  = 1  + (0.654 + 0.755i)3-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.959 + 0.281i)25-s + (−0.841 + 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.178 + 0.983i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.178 + 0.983i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $0.178 + 0.983i$
motivic weight  =  \(0\)
character  :  $\chi_{92} (3, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 92,\ (1:\ ),\ 0.178 + 0.983i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.482850506 + 1.237901014i$
$L(\frac12,\chi)$  $\approx$  $1.482850506 + 1.237901014i$
$L(\chi,1)$  $\approx$  1.234806650 + 0.4499696598i
$L(1,\chi)$  $\approx$  1.234806650 + 0.4499696598i

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

−30.15933219976261892424418740801, −29.28917705495921396471488379712, −27.53528142407963806451377639232, −26.569804503705975491826293658526, −25.66307792313923449113284150422, −24.82085710264608321718857401773, −23.33541636677874739860753306477, −22.83427642905755531234694153647, −21.290968109403925262026622358166, −19.89479795551011078888310339549, −19.3357717421579243683544038596, −18.191789182250434912214358466146, −17.145642601123555369697815278124, −15.54008190068117279917674771839, −14.33999362939158907279194055617, −13.66622798632950824856083043972, −12.35810584695332609437034077287, −10.98649715865278574176919835980, −9.76026694773889611112726002428, −8.23627068585385023637207486259, −7.10226991540507293088382562564, −6.2674756651481775361984182590, −3.81891792577177107555900738972, −2.82210653992362497672918025882, −0.87717114495869139512230307560, 1.805158219570453969461263840657, 3.60894831769389012065628684042, 4.72315324013003901460992092215, 6.19834667240852470333108899338, 8.23517664762689935531910975023, 9.00871698108334970166974634900, 9.93785120334933841140141319823, 11.68316401139650586337854857042, 12.71523947184592322121241527124, 14.11785242625286632214825665589, 15.18949644898338970558069097574, 16.243866545451809950299219681506, 17.028458924110294085872123181419, 18.94401115026260663967363110231, 19.66177319849228976806849141092, 20.9304729573463058212422133580, 21.5512824023151398678422977169, 22.80648741837304741462597929198, 24.23955705088420212328005564985, 25.28179651644809580962494275312, 25.91921595603587492949505833774, 27.54415688819423674930707557013, 27.917683693840366426421659327161, 29.0568040402808373600245435355, 30.66504213941491342840255021259

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