Properties

Degree 1
Conductor 41
Sign $-0.661 + 0.749i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.951 + 0.309i)2-s + (−0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (−0.891 + 0.453i)6-s + (−0.891 − 0.453i)7-s + (0.587 + 0.809i)8-s i·9-s + (−0.809 + 0.587i)10-s + (−0.156 + 0.987i)11-s + (−0.987 + 0.156i)12-s + (0.453 + 0.891i)13-s + (−0.707 − 0.707i)14-s + (−0.156 − 0.987i)15-s + (0.309 + 0.951i)16-s + (0.987 + 0.156i)17-s + ⋯
L(s,χ)  = 1  + (0.951 + 0.309i)2-s + (−0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (−0.891 + 0.453i)6-s + (−0.891 − 0.453i)7-s + (0.587 + 0.809i)8-s i·9-s + (−0.809 + 0.587i)10-s + (−0.156 + 0.987i)11-s + (−0.987 + 0.156i)12-s + (0.453 + 0.891i)13-s + (−0.707 − 0.707i)14-s + (−0.156 − 0.987i)15-s + (0.309 + 0.951i)16-s + (0.987 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.661 + 0.749i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.661 + 0.749i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $-0.661 + 0.749i$
motivic weight  =  \(0\)
character  :  $\chi_{41} (26, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 41,\ (1:\ ),\ -0.661 + 0.749i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6729055799 + 1.491983522i$
$L(\frac12,\chi)$  $\approx$  $0.6729055799 + 1.491983522i$
$L(\chi,1)$  $\approx$  1.023766646 + 0.8004843659i
$L(1,\chi)$  $\approx$  1.023766646 + 0.8004843659i

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

−34.40112538164095658340871852742, −32.6853960812580782577032783529, −31.8988087549391011015803577952, −30.69991884172316337822247624571, −29.458303008898304810648104646119, −28.68081401588287377498735593786, −27.61533598573115453515537746090, −25.22959769037596414216454221872, −24.402826393907151675157399305323, −23.23826233711883285967640511069, −22.49840649978178221280150227820, −21.00385847079491592661595880903, −19.587920599331593496894609853535, −18.655711274187246033663991072, −16.514341942725312675384858610900, −15.82367041726060293470669545646, −13.783852158985861736252823392286, −12.58895093243244807943044395317, −11.97528660007914481180979820906, −10.42298347794919516037006233091, −8.08331957482792937141525805832, −6.26470247728448078138696245484, −5.23691293896699851396966147477, −3.240198581243420492737837083632, −0.874179625199304787850445905404, 3.29483036967315025829631670937, 4.45936508538561745617847093798, 6.23598321994723861335216780823, 7.29653243598609805153400975353, 9.90994124717835734784792967513, 11.25069234649182643277702543942, 12.33774713033189771653456639865, 14.04749245645968644191277009762, 15.43049324051335569871295678794, 16.134934394061784444567461934704, 17.540726594795404805124350945848, 19.475971553221859662940446500859, 20.93313596344063023802369098883, 22.16013592379341195103693129047, 23.04719070115530769550936848735, 23.64588223566431347508941500633, 25.80340858541707562041189675419, 26.41581777026054848137726805329, 28.09730181415714847078202563135, 29.35263563907020470536416551480, 30.49051818434461585650639293679, 31.71383052920746522826744682494, 32.89011131722217324851166934882, 33.71424207290313302459421022631, 34.68997971595070885116514726056

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