Properties

Degree 1
Conductor $ 2^{3} \cdot 5 $
Sign $-0.525 - 0.850i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  i·3-s i·7-s − 9-s − 11-s i·13-s i·17-s + 19-s − 21-s + i·23-s + i·27-s + 29-s + 31-s + i·33-s + i·37-s − 39-s + ⋯
L(s,χ)  = 1  i·3-s i·7-s − 9-s − 11-s i·13-s i·17-s + 19-s − 21-s + i·23-s + i·27-s + 29-s + 31-s + i·33-s + i·37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.525 - 0.850i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.525 - 0.850i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(40\)    =    \(2^{3} \cdot 5\)
\( \varepsilon \)  =  $-0.525 - 0.850i$
motivic weight  =  \(0\)
character  :  $\chi_{40} (13, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 40,\ (1:\ ),\ -0.525 - 0.850i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5840185059 - 1.047498216i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5840185059 - 1.047498216i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8450865538 - 0.5222922137i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8450865538 - 0.5222922137i\)

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

−34.889410722913579019042779215255, −34.00593978863417924291162844420, −32.86231382272903645228213911845, −31.67062597000032905452552710638, −30.87016611756245345183516519694, −28.75483491129007218737184032280, −28.294360278563347705201931249151, −26.74136643289033570614674792378, −25.92992580013884052785017865113, −24.44019338983484927528478730337, −22.93786179582473127180677905639, −21.66654677955369228655182343419, −20.95428797352835588493580628252, −19.35445000819576732156807553178, −17.95182105004652510585130909180, −16.35256477930716606495463617653, −15.44477138266542007247681781931, −14.179957072571212635060024343162, −12.314118211724954003592304568031, −10.909635654089499426674194736919, −9.5392668092564573127924385132, −8.29247179462172609731476339067, −5.98776315237030887203366066327, −4.5862074698134147263949347033, −2.712571261673437121457390540174, 0.78175758422594404739951629151, 2.96145731604313918366439218618, 5.31837883635310062185777995034, 7.097471960447107766621899830927, 8.06518186518042676061520118383, 10.13039025039574289642412239789, 11.633914964105169352606859301659, 13.141235995668734664727808836599, 13.94588662492370748761203109124, 15.79427994969970989519583146281, 17.38576376059781498840809763208, 18.29799312238130571037906726382, 19.74478701066218196150919767223, 20.695607168607225609497625932771, 22.69999016720834375730312406956, 23.53597150240823310138026442521, 24.722920665161044269632960455613, 25.90667840123069701772422442237, 27.174906699397835017340244911219, 28.83127379121171043186545706856, 29.66415484553858424483262837154, 30.720036159678105293485643075963, 31.88487219300669509430517189192, 33.368390162792623722285111405705, 34.512665885422244155468985822560

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