Properties

Degree 1
Conductor 23
Sign $0.854 + 0.519i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−0.654 − 0.755i)11-s + (−0.654 − 0.755i)12-s + (−0.959 − 0.281i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + ⋯
L(s,χ)  = 1  + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−0.654 − 0.755i)11-s + (−0.654 − 0.755i)12-s + (−0.959 − 0.281i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.854 + 0.519i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.854 + 0.519i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $0.854 + 0.519i$
motivic weight  =  \(0\)
character  :  $\chi_{23} (4, \cdot )$
Sato-Tate  :  $\mu(11)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 23,\ (0:\ ),\ 0.854 + 0.519i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5936953919 + 0.1662737264i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5936953919 + 0.1662737264i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8044174782 + 0.1800159172i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8044174782 + 0.1800159172i\)

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

−38.82904969900439216797491855283, −37.635050660222910611763365632394, −36.27341873101567924360620203011, −35.938907075447944483317436388978, −33.703928987552825404236293326258, −32.1091735901850118059187416565, −31.32056718767089627329744047203, −29.53655072465957060785288875489, −28.50879775493850333463910445784, −27.13223980683776795140294166042, −25.9574329561884069176049822057, −24.93649520853625830521165936466, −22.4678459530138552535662630855, −20.977937859874896897254665650164, −20.21715299352348244292115288302, −18.98899807308690491928557869720, −17.07638010143655944598506026097, −15.89863210493409896148305765933, −13.61944875373012828405910231195, −12.425109771331570931137449711022, −10.03959677418238791346100433673, −9.39325673406057682914387718310, −7.720799325920089958960128054403, −4.432117621973441540755241160510, −2.47039178966832528664961712539, 2.695063685459656135537545911260, 6.13271706638252198914368019481, 7.386736091739146298255844220707, 9.0236073970210270980590936589, 10.40381591166334108468080526150, 13.12631838826265572682657003615, 14.4875947273727442066818885853, 15.65730440476089251256326484275, 17.60123837809627875346084807219, 18.86079323503127356346653185205, 19.63149511760766360944426352873, 21.899117064346872748693005649887, 23.61823174774936248044005691048, 24.959998697416946641111554588972, 26.03410951890411644829689228165, 26.68424173530906038326597384210, 28.79736871063084622912026153343, 29.90051658251202674860051032003, 31.68697678664476224776710908677, 32.686616841364382480753785584274, 34.39659421649262485665831178230, 35.19931983131626181222038711038, 36.73684444152570151931867284665, 37.374582052745730403293364755146, 38.700408365765894581394546516791

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