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Properties

Label 1-41-41.26-r1-0-0
Degree $1$
Conductor $41$
Sign $-0.661 + 0.749i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.661 + 0.749i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ -0.661 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6729055799 + 1.491983522i\)
\(L(\frac12)\) \(\approx\) \(0.6729055799 + 1.491983522i\)
\(L(1)\) \(\approx\) \(1.023766646 + 0.8004843659i\)
\(L(1)\) \(\approx\) \(1.023766646 + 0.8004843659i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (-0.891 - 0.453i)T \)
11 \( 1 + (-0.156 + 0.987i)T \)
13 \( 1 + (0.453 + 0.891i)T \)
17 \( 1 + (0.987 + 0.156i)T \)
19 \( 1 + (0.453 - 0.891i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.987 - 0.156i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (-0.987 + 0.156i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (0.156 + 0.987i)T \)
71 \( 1 + (0.156 - 0.987i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 + (-0.156 - 0.987i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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