Properties

Degree $1$
Conductor $2015$
Sign $0.902 - 0.430i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 − 0.5i)6-s + (0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + 16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯
L(s,χ)  = 1  + 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 − 0.5i)6-s + (0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + 16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.902 - 0.430i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.902 - 0.430i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.902 - 0.430i$
Motivic weight: \(0\)
Character: $\chi_{2015} (57, \cdot )$
Sato-Tate group: $\mu(12)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.902 - 0.430i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(9.072637589 - 2.050595953i\)
\(L(\frac12,\chi)\) \(\approx\) \(9.072637589 - 2.050595953i\)
\(L(\chi,1)\) \(\approx\) \(3.318515418 - 0.4291659484i\)
\(L(1,\chi)\) \(\approx\) \(3.318515418 - 0.4291659484i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.929246290148147760794226066695, −19.5038778376312411240117108089, −18.61881705267038083148142645129, −17.27105799611805379413830313987, −16.73695672016339377766618281946, −15.980792136141388358005880609413, −15.2654924182749084527469775192, −14.44620258382145718309834029364, −13.97428328248451015822525968892, −13.61877418778859463141591420123, −12.54054237354324777261976511154, −11.72948027777372573382594952248, −10.96947872168426345437502562127, −10.209971568851367927924365399584, −9.53933082456103301039776258134, −8.321476329074753719767020937140, −7.76207165629033147106990820061, −6.99718185573328697986621091811, −6.00068093154423645593807424715, −5.05354276712438822691461750614, −4.35630878197004028739100091836, −3.43522188742516738717616122190, −3.23729718302029803498137535657, −1.73520303393874812314595549009, −1.218549329999223683480586718872, 1.04894096451542021816477514719, 1.80116637691260646142325320965, 2.71691328024423345836430525345, 3.26402771663509263385144784650, 4.37123096713819569086449438471, 5.03152072281135144250189470946, 6.08594885611839244039901557044, 6.76938150229113980996294149765, 7.57311764580794845904922880707, 8.27879772615355161615175410575, 9.21736091058637020768426922324, 9.923607586531935852414625354895, 11.12884904770516554114461130214, 12.009985763116697522177615769, 12.26895184371581802156320871815, 13.11139725007165457312254733344, 14.106916745831243410092278620769, 14.38412729944621572120378078555, 15.049611650170864587269855421502, 15.78720213420641987189235396180, 16.54878859727115397149194743767, 17.712555611159175726464027800275, 18.28491438653218990428803176213, 19.24251954178006018131039761415, 19.78876835619834116337074856638

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