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Properties

Label 1-92-92.31-r1-0-0
Degree $1$
Conductor $92$
Sign $0.178 - 0.983i$
Analytic cond. $9.88677$
Root an. cond. $9.88677$
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  + (0.654 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.178 - 0.983i$
Analytic conductor: \(9.88677\)
Root analytic conductor: \(9.88677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (1:\ ),\ 0.178 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.482850506 - 1.237901014i\)
\(L(\frac12)\) \(\approx\) \(1.482850506 - 1.237901014i\)
\(L(1)\) \(\approx\) \(1.234806650 - 0.4499696598i\)
\(L(1)\) \(\approx\) \(1.234806650 - 0.4499696598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.654 - 0.755i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (-0.142 - 0.989i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 - T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

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

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