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Properties

Label 1-41-41.28-r1-0-0
Degree $1$
Conductor $41$
Sign $-0.228 + 0.973i$
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  + (0.587 + 0.809i)2-s + (0.707 + 0.707i)3-s + (−0.309 + 0.951i)4-s + (0.951 + 0.309i)5-s + (−0.156 + 0.987i)6-s + (−0.156 − 0.987i)7-s + (−0.951 + 0.309i)8-s + i·9-s + (0.309 + 0.951i)10-s + (0.453 − 0.891i)11-s + (−0.891 + 0.453i)12-s + (−0.987 − 0.156i)13-s + (0.707 − 0.707i)14-s + (0.453 + 0.891i)15-s + (−0.809 − 0.587i)16-s + (0.891 + 0.453i)17-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.707 + 0.707i)3-s + (−0.309 + 0.951i)4-s + (0.951 + 0.309i)5-s + (−0.156 + 0.987i)6-s + (−0.156 − 0.987i)7-s + (−0.951 + 0.309i)8-s + i·9-s + (0.309 + 0.951i)10-s + (0.453 − 0.891i)11-s + (−0.891 + 0.453i)12-s + (−0.987 − 0.156i)13-s + (0.707 − 0.707i)14-s + (0.453 + 0.891i)15-s + (−0.809 − 0.587i)16-s + (0.891 + 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.228 + 0.973i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ -0.228 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.570716065 + 1.982039782i\)
\(L(\frac12)\) \(\approx\) \(1.570716065 + 1.982039782i\)
\(L(1)\) \(\approx\) \(1.452804593 + 1.114671296i\)
\(L(1)\) \(\approx\) \(1.452804593 + 1.114671296i\)

Euler product

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

−34.197158557060623473772740328618, −32.629445376657949226789228676460, −31.805001037979794943903604709215, −30.79074445437229591162652202513, −29.61592123492115326208788431231, −28.85816157575666729425208724196, −27.51767935216950258284310375231, −25.456946076109048328831399745768, −24.87345652742755983481181457849, −23.47228349927807842680307967447, −21.94631223158596377210104728741, −21.00082064045830516776659731997, −19.77257706605023346748772617837, −18.712281595460906208190854616651, −17.50907556297056581254913560687, −15.05104321582336742796695361485, −14.15031711313357200026494310993, −12.76726550614957559548941324263, −12.1130185448413933660405472985, −9.85139962292055437059502507244, −8.95352462567187507425350069846, −6.63213244811130860440130627372, −5.06359892797314647829820109254, −2.817653428584656925236350726303, −1.66914631617698378760206868839, 2.95909149414783302258387143758, 4.436559627804492888405206485303, 6.10265265287133442535077538189, 7.7087857650828650168384003102, 9.25479772579750361960605368941, 10.6118892983353565584003562852, 12.98041660156207099618319267555, 14.13073013406655189309846055288, 14.76518844354503528328120161563, 16.53156006735854602328917409473, 17.199880140666461416988891797784, 19.241667951987845230366258909078, 20.910111691049616261153033230501, 21.710317605989105955558236557544, 22.839112048069098682824138749379, 24.45647919862998073313048741708, 25.48558984512545688404173237536, 26.43580576020041911315940732984, 27.27377209907068440813909275795, 29.54096698115388632515263158178, 30.377778519836176464964324349162, 32.000233759695170456693017947283, 32.58469537248153062875023733837, 33.522633394099052618542179210216, 34.57098790426990201141850914392

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