Properties

Degree 1
Conductor $ 2^{2} \cdot 97 $
Sign $-0.746 + 0.665i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.382 + 0.923i)3-s + (−0.195 + 0.980i)5-s + (0.195 + 0.980i)7-s + (−0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.980 + 0.195i)13-s + (−0.980 + 0.195i)15-s + (0.980 + 0.195i)17-s + (−0.195 + 0.980i)19-s + (−0.831 + 0.555i)21-s + (0.555 + 0.831i)23-s + (−0.923 − 0.382i)25-s + (−0.923 − 0.382i)27-s + (−0.555 − 0.831i)29-s + (−0.923 + 0.382i)31-s + ⋯
L(s,χ)  = 1  + (0.382 + 0.923i)3-s + (−0.195 + 0.980i)5-s + (0.195 + 0.980i)7-s + (−0.707 + 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.980 + 0.195i)13-s + (−0.980 + 0.195i)15-s + (0.980 + 0.195i)17-s + (−0.195 + 0.980i)19-s + (−0.831 + 0.555i)21-s + (0.555 + 0.831i)23-s + (−0.923 − 0.382i)25-s + (−0.923 − 0.382i)27-s + (−0.555 − 0.831i)29-s + (−0.923 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.746 + 0.665i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.746 + 0.665i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(388\)    =    \(2^{2} \cdot 97\)
\( \varepsilon \)  =  $-0.746 + 0.665i$
motivic weight  =  \(0\)
character  :  $\chi_{388} (131, \cdot )$
Sato-Tate  :  $\mu(32)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 388,\ (0:\ ),\ -0.746 + 0.665i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4779962024 + 1.253871173i$
$L(\frac12,\chi)$  $\approx$  $0.4779962024 + 1.253871173i$
$L(\chi,1)$  $\approx$  0.9155920387 + 0.6896338542i
$L(1,\chi)$  $\approx$  0.9155920387 + 0.6896338542i

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

−23.99311665674876596516952408593, −23.48560539212476870687104440578, −22.849757388598917689022169160269, −21.16570631971381021726260395714, −20.3096806539935332574750082167, −20.133317561092512432896264226762, −18.89283987278813157080905070547, −18.0200377598080020556317947033, −17.1605695949893361597770690548, −16.382289176700823437805397524679, −15.18314615323046308959660445004, −14.2125967897107383130259019848, −13.14506513462210620054999420791, −12.84353108025847114691725005334, −11.72512801213735610913672978302, −10.661069279664996934510568769070, −9.38021498463565759083288679009, −8.466269775529752724126263282025, −7.60575358105353807345639397390, −6.86216793508697930255906553803, −5.48442469740134127480720071701, −4.40416671985284525527731849507, −3.23640992404060991092214219821, −1.73173970643785589092326518813, −0.790901093559745529471104436869, 2.02908779421757074676342483285, 3.25476434195179109187339103395, 3.76246124659393391508364525886, 5.47070966901575741434974221822, 5.96852494302959502624386124050, 7.61335198078831413273076379136, 8.46439982193706926548392813092, 9.349278828021588620765153271276, 10.49982301774948571860407835895, 11.094144735101684985863145565492, 12.031696343944441992191760650035, 13.51975994936658469599767214505, 14.35472920567511747316120074884, 15.09352584819204389610694251523, 15.84262707819819299836201235727, 16.61836396871417621679958794614, 17.96911820520331282025036682884, 18.92545185006624550160670532423, 19.268773855260343978974400172381, 20.947230443821962373366516780946, 21.16858516467337174726178831554, 22.11989112138775346346251903750, 22.89309267621767837610468391197, 23.81903901266557640382011417614, 25.20425886896187135583965744299

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