Properties

Label 1-41-41.8-r0-0-0
Degree $1$
Conductor $41$
Sign $0.936 - 0.350i$
Analytic cond. $0.190403$
Root an. cond. $0.190403$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.936 - 0.350i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 0.936 - 0.350i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7435439189 - 0.1345510543i\)
\(L(\frac12)\) \(\approx\) \(0.7435439189 - 0.1345510543i\)
\(L(1)\) \(\approx\) \(0.8931869235 - 0.1502334416i\)
\(L(1)\) \(\approx\) \(0.8931869235 - 0.1502334416i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−34.84492989689333102317541957834, −33.95613810425839318272241592875, −33.10146037457388053761408012685, −31.45628429844859988943835137431, −30.306550888717162240916960185701, −29.20897375568625454194628132740, −27.71017837077292800105384028705, −26.441417119097055665488293433323, −25.1670951451865974099914391556, −24.59798499334746443881884336054, −23.257395760578834406164080694836, −22.18543097112789863657222626694, −20.151327344362252934944742615763, −18.65591191911217618291139135153, −17.69059005594408796905291170035, −17.06184641998449843548822419318, −14.72664683044227520258768975765, −14.21333029945417462397308716551, −12.71936315991670235423090371567, −10.73621239579423010521725766665, −9.07324282943668093729834757162, −7.489132257117115969338429871623, −6.622952248093256589101809671853, −5.02594343695939778134044442951, −1.89908354839906079690618186824, 2.095984176596019545908096785206, 4.14555596102408205293084735003, 5.41753766607884998636530345220, 8.50942939244626961642751580413, 9.337749704534744348573341967730, 10.70486116927952795098884052571, 11.83901716729304691336568531210, 13.54315473037126072317189103010, 14.8402301159347132340257963630, 16.92082342003892062103749149686, 17.42856347690545647644025557591, 19.29178886821563968130562691150, 20.56407650283102621488637791309, 21.56053533825359507522446065814, 22.01850755786501914209671214564, 24.09807514478065329394380530916, 25.67540043635038621557104725823, 26.977041435894928190630199617386, 27.801099391160869055849538294056, 28.79980384276252059826318779723, 29.95171962965405071761449567937, 31.471710979620936211167202800411, 32.29744784477114078932880658513, 33.59011263178657020691959958479, 34.89976552232428313903304403539

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