Properties

Degree 1
Conductor 43
Sign $-0.863 + 0.503i$
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 + (−0.623 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + 6-s − 7-s + (0.900 + 0.433i)8-s + (−0.222 + 0.974i)9-s + (−0.900 + 0.433i)10-s + (−0.222 + 0.974i)11-s + (−0.623 + 0.781i)12-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.222 − 0.974i)15-s + (−0.900 + 0.433i)16-s + (−0.900 + 0.433i)17-s + ⋯
L(s,χ)  = 1  + (−0.623 + 0.781i)2-s + (−0.623 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + 6-s − 7-s + (0.900 + 0.433i)8-s + (−0.222 + 0.974i)9-s + (−0.900 + 0.433i)10-s + (−0.222 + 0.974i)11-s + (−0.623 + 0.781i)12-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.222 − 0.974i)15-s + (−0.900 + 0.433i)16-s + (−0.900 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.863 + 0.503i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.863 + 0.503i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.863 + 0.503i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (2, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 43,\ (1:\ ),\ -0.863 + 0.503i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1054053054 + 0.3900837046i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1054053054 + 0.3900837046i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4958056742 + 0.1921787255i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4958056742 + 0.1921787255i\)

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

−34.02072831906720117119188726072, −32.55301559437363267207950878134, −31.748361158531520564457942223679, −29.74789925566173855084644265489, −28.91720261488249095978811622513, −28.378507997918933770391704876796, −26.76498646845027129459169482019, −26.15738433794192657087987819016, −24.49052413504911601527412476861, −22.44939269611324043311414348206, −21.80737970417227387660387986782, −20.7324502335511626413725294173, −19.421353368288826773282091725477, −17.89866973599801925560993613713, −16.84494649510486752742849610887, −15.98493974903939487859238296959, −13.644576978521201121266744092102, −12.3652769717193273741510771318, −10.93791455220880097483534998869, −9.74600042762669450244381679292, −8.96694976561699129794887034542, −6.48388230773465929620484304193, −4.69301881409986213753699386326, −2.81555328701758415957416244905, −0.31180566149356018981619117229, 1.98297127705129717614369871831, 5.39194248130293473863232787345, 6.50959148719987269558665768959, 7.51379086614318658440245752914, 9.5490711137965876681680040380, 10.54329990297302720110742519835, 12.61245735566498458451960456473, 13.8124595074614612598439226790, 15.3458268222217986882155934803, 16.907169104453699209045057222245, 17.6819924922444588473096699976, 18.716261231664026383874665821733, 19.88649301482045938640359632710, 22.25287880498495567654870976688, 22.9577980006859578515331499926, 24.42517123594303822440225814080, 25.37173837210533963856873936405, 26.23126371977121243914640707754, 27.902458316237283635544390450097, 29.01493173255697605601444410012, 29.64714306926130996129436246175, 31.46030069972594153070404267625, 33.05864393992721237743127711601, 33.72960928722102665062275818000, 34.92536792258148701314569426534

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