Properties

Degree 1
Conductor 43
Sign $0.157 - 0.987i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.157 - 0.987i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.157 - 0.987i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.157 - 0.987i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (31, \cdot )$
Sato-Tate  :  $\mu(21)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 43,\ (0:\ ),\ 0.157 - 0.987i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8484034959 - 0.7238840254i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8484034959 - 0.7238840254i\)
\(L(\chi,1)\)  \(\approx\)  \(1.112498308 - 0.6645510864i\)
\(L(1,\chi)\)  \(\approx\)  \(1.112498308 - 0.6645510864i\)

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

−34.69979799594470525653595015858, −33.25045292441879439739786706462, −32.61317426167013997818981457212, −31.892278992719177793450561360749, −30.65884992850541274654970825995, −29.02580347577660053987162167880, −27.55475190235180512449230636826, −26.36449340857793578122572297926, −25.578094084909520225220224742883, −24.10782006248384739685510918438, −23.157149554445700135315192191643, −21.58273481291060429496182689135, −20.92678237023006622577162062553, −19.5105682108041306237690429218, −17.08101458144328065794852581311, −16.455577161820967170578301065722, −15.41866174627825352520825382460, −13.84923045080689374170613676172, −13.05058458149009305807353968196, −11.039233093085265251085647564850, −9.17472698544769065718607396026, −8.16817372721036357906181020443, −6.128659999511946818588170357736, −4.610412422979667021858289768281, −3.51552786162632649791928835426, 2.16398920530801459601881800336, 3.27868348410806140491048592593, 5.73565563183579492499247620217, 7.01750528120982129188148214138, 9.08328765548757633589829903603, 10.6613005523596263382415555360, 12.123818715463291868417699726259, 13.065517530267463491510043392940, 14.402847916368514632500808378035, 15.35958072296610883228430957503, 18.14094713679934933557411899508, 18.617633029667930739361682308821, 19.8438421427573398141947167728, 21.1195183750048577971123975465, 22.69878256377497212285767668381, 23.17595903950390638284750289562, 24.94187061481132174282277021702, 25.77122819197733464618868332322, 27.62850515140048283031776945268, 28.9792454234269828454880747914, 29.79998754436902218527599131966, 30.94184039025489551732138694374, 31.45527662953620240033473411192, 33.02772553046591218557131881081, 34.39118067980674391983820444919

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