Properties

Degree 1
Conductor 43
Sign $0.0861 - 0.996i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0861 - 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0861 - 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.0861 - 0.996i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (7, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 43,\ (1:\ ),\ 0.0861 - 0.996i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9197477990 - 0.8436709768i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9197477990 - 0.8436709768i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8679828422 - 0.4058429752i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8679828422 - 0.4058429752i\)

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

−34.45786116028905241653470247152, −33.449959835310220127057985134049, −32.85033601298427520494306920868, −30.82964101792448072500199269662, −29.89805525780254720101460713335, −28.55913474612112704110269309246, −27.104367014013485234884802833643, −26.62873154441361867637989874986, −25.61523355235269918386877170955, −24.3377335335006632025507789482, −22.34312742129952092821767183731, −21.19768794114394319455197829469, −20.06701402255949239176438154445, −18.98047379402767906970900106782, −17.41119871258341444433738695709, −16.56181692370333606205782380282, −14.88582026183279639560637185018, −14.11651098976765992036488623780, −11.45698066352487407606972334062, −10.36911215794911280288166030177, −9.43800991223500845865315463337, −7.89943696573456148333166205595, −6.40702451959329461376804207481, −3.9370644710261553809570052198, −2.06110075350992274517754634234, 1.09580141148963236096486757852, 2.52271332353344334609617653779, 5.646701597283202001520821953513, 7.274221837677492256275699867459, 8.686153496405728458400500866573, 9.38259720648212090240080904773, 11.60641861797986386818867806824, 12.6126904840729971507181535187, 14.32354414208736381328855795403, 15.80672083209480138750179692690, 17.523393775315170123987799323931, 18.01182952892485477824263365019, 19.62436074727165960834586876179, 20.30192963838427358199968174424, 21.75834566481149942836236027872, 24.06758745889883642503268159709, 24.903929149817166522275972064274, 25.396383436861469299782498533293, 27.11634759005845422161916774886, 28.21226763417084744022917991808, 29.28580260602621801917013327816, 30.27256313198806360753900635578, 31.65968148928727128084356556708, 32.99439858440759570823697586390, 34.56341959616186880179420607754

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