Properties

Degree 1
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.989 - 0.141i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.654 − 0.755i)5-s + (−0.0475 + 0.998i)7-s + (0.981 + 0.189i)11-s + (0.928 + 0.371i)13-s + (−0.580 − 0.814i)17-s + (−0.0475 − 0.998i)19-s + (0.723 − 0.690i)23-s + (−0.142 − 0.989i)25-s + (0.5 + 0.866i)29-s + (−0.928 + 0.371i)31-s + (0.723 + 0.690i)35-s + (−0.5 + 0.866i)37-s + (0.995 − 0.0950i)41-s + (−0.415 + 0.909i)43-s + (0.235 + 0.971i)47-s + ⋯
L(s,χ)  = 1  + (0.654 − 0.755i)5-s + (−0.0475 + 0.998i)7-s + (0.981 + 0.189i)11-s + (0.928 + 0.371i)13-s + (−0.580 − 0.814i)17-s + (−0.0475 − 0.998i)19-s + (0.723 − 0.690i)23-s + (−0.142 − 0.989i)25-s + (0.5 + 0.866i)29-s + (−0.928 + 0.371i)31-s + (0.723 + 0.690i)35-s + (−0.5 + 0.866i)37-s + (0.995 − 0.0950i)41-s + (−0.415 + 0.909i)43-s + (0.235 + 0.971i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.989 - 0.141i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.989 - 0.141i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.989 - 0.141i$
motivic weight  =  \(0\)
character  :  $\chi_{804} (23, \cdot )$
Sato-Tate  :  $\mu(66)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 804,\ (0:\ ),\ 0.989 - 0.141i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.806614883 - 0.1284550877i$
$L(\frac12,\chi)$  $\approx$  $1.806614883 - 0.1284550877i$
$L(\chi,1)$  $\approx$  1.301210940 - 0.05200355248i
$L(1,\chi)$  $\approx$  1.301210940 - 0.05200355248i

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

−22.34118772756379548454136623213, −21.466301895968181873468893628197, −20.74820546123183272453057314160, −19.82428574847898018090958083243, −19.09870110685011868562596857856, −18.222583427245977898185823470394, −17.3055858718773898662676592601, −16.93629418741630555839237151711, −15.77664161185615994720461549784, −14.84083314057473760349613636234, −14.07187060423365281215563260073, −13.49138585778315490495580630654, −12.61913046806790301148625435802, −11.28573203993803560458066367831, −10.79264608468882729191579595369, −9.97109614547176930545038252421, −9.082081915369725925486762403379, −8.0196533062810383867431576508, −7.00454467923052928889038466604, −6.315330459642823802845694994215, −5.51474715980122320371553635055, −3.90955499867455301934435187659, −3.593168233138516963737050294472, −2.09616259504457319230403369658, −1.12502135193626299799691265521, 1.08156581489467879282492188757, 2.07193229238483226647381384997, 3.125378749654730929577529545451, 4.51207082791464729027162729891, 5.15446014181220927682513195796, 6.26526320375052489115181998211, 6.83465721153085298743861402027, 8.392358795014022392587840055587, 9.11168859817208445189042850454, 9.37093694267531784691035318304, 10.81906486280679969440871489748, 11.62415333410760408119994535218, 12.48559325414921017829722313484, 13.20288398307059262038734221743, 14.07430279097665594649973282824, 14.91423267268776913635543199211, 15.97205912023073744055332971151, 16.447574592600161265103268342477, 17.58166792804893684609657679140, 18.05441975748938566181479245179, 19.02581812748529622192238436278, 19.92865136060107883002055861941, 20.680798705045287605637044780603, 21.47631415452875008678648874469, 22.11041015775104196760368634895

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