Properties

Degree 1
Conductor 41
Sign $0.936 + 0.350i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.936 + 0.350i$
motivic weight  =  \(0\)
character  :  $\chi_{41} (36, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 41,\ (0:\ ),\ 0.936 + 0.350i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7435439189 + 0.1345510543i$
$L(\frac12,\chi)$  $\approx$  $0.7435439189 + 0.1345510543i$
$L(\chi,1)$  $\approx$  0.8931869235 + 0.1502334416i
$L(1,\chi)$  $\approx$  0.8931869235 + 0.1502334416i

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

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

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