Properties

Degree 1
Conductor 43
Sign $-0.974 - 0.224i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.222 + 0.974i)2-s + (−0.733 + 0.680i)3-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (0.623 − 0.781i)8-s + (0.0747 − 0.997i)9-s + (0.365 − 0.930i)10-s + (−0.900 + 0.433i)11-s + (0.955 − 0.294i)12-s + (0.365 + 0.930i)13-s + (−0.733 − 0.680i)14-s + (0.826 − 0.563i)15-s + (0.623 + 0.781i)16-s + (−0.988 + 0.149i)17-s + ⋯
L(s,χ)  = 1  + (−0.222 + 0.974i)2-s + (−0.733 + 0.680i)3-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (0.623 − 0.781i)8-s + (0.0747 − 0.997i)9-s + (0.365 − 0.930i)10-s + (−0.900 + 0.433i)11-s + (0.955 − 0.294i)12-s + (0.365 + 0.930i)13-s + (−0.733 − 0.680i)14-s + (0.826 − 0.563i)15-s + (0.623 + 0.781i)16-s + (−0.988 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.974 - 0.224i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.974 - 0.224i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.974 - 0.224i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (23, \cdot )$
Sato-Tate  :  $\mu(21)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 43,\ (0:\ ),\ -0.974 - 0.224i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.03489669618 + 0.3066401376i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.03489669618 + 0.3066401376i\)
\(L(\chi,1)\)  \(\approx\)  \(0.3069948015 + 0.3691771416i\)
\(L(1,\chi)\)  \(\approx\)  \(0.3069948015 + 0.3691771416i\)

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

−34.48528394990567031650749671359, −32.68074275174775144446186251651, −31.217864601348866161350787844806, −30.324461576019623745111001515582, −29.336825209211590434891155212228, −28.36549029129074765978383429226, −27.19694857740121948591766800711, −26.14564916342342734533114042328, −24.05248676612151045609989013852, −23.06168325732325853828218242497, −22.30372646986335564071857275906, −20.44747987938863179700395734040, −19.443133613421742782324470384772, −18.44000409440558201877045208917, −17.24429786828670950180094466176, −15.838543155392810162598078154329, −13.48805972583942463148234000946, −12.71265420851421804403822927401, −11.20157876611161662444454862370, −10.56912856909675178424211327165, −8.34357261680893085057755851413, −7.05881320317048283816271504797, −4.82148810372987725658039663499, −3.0485822611409087390100742034, −0.529152786233597396561026093606, 4.034754976978269690880841748404, 5.369142871812428125291054494866, 6.78914749528408869797025572592, 8.497237356435967123512938854303, 9.771338142306875066036435570211, 11.42788122230091777212412363674, 12.88828436974980551642735455272, 15.03844779621102605322336903782, 15.73089390545477927478650708991, 16.6094291134842821002334728310, 18.11312838127546948339321421250, 19.19917715700349906741026053283, 21.129479441409714658083579623080, 22.5944303719705488862379795798, 23.29318266455219131472296751558, 24.467294216428494220755867030340, 26.06376796851191892237217302748, 26.92165968636260530587315280252, 28.16061115118675539547319554939, 28.723211101021416582206269032386, 31.282189756132228631398801866146, 31.77981033617351490970883782818, 33.27126773490391371408950117183, 34.056154859023755683525699800480, 35.20216715465300766704862070626

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