Properties

Degree 1
Conductor 311
Sign $0.945 + 0.324i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.440 − 0.897i)2-s + (−0.440 + 0.897i)3-s + (−0.612 + 0.790i)4-s + (−0.874 − 0.485i)5-s + 6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (−0.612 − 0.790i)9-s + (−0.0506 + 0.998i)10-s + (0.528 − 0.848i)11-s + (−0.440 − 0.897i)12-s + (−0.612 − 0.790i)13-s + (0.347 − 0.937i)14-s + (0.820 − 0.571i)15-s + (−0.250 − 0.968i)16-s + (−0.0506 + 0.998i)17-s + ⋯
L(s,χ)  = 1  + (−0.440 − 0.897i)2-s + (−0.440 + 0.897i)3-s + (−0.612 + 0.790i)4-s + (−0.874 − 0.485i)5-s + 6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (−0.612 − 0.790i)9-s + (−0.0506 + 0.998i)10-s + (0.528 − 0.848i)11-s + (−0.440 − 0.897i)12-s + (−0.612 − 0.790i)13-s + (0.347 − 0.937i)14-s + (0.820 − 0.571i)15-s + (−0.250 − 0.968i)16-s + (−0.0506 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.945 + 0.324i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.945 + 0.324i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $0.945 + 0.324i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (13, \cdot )$
Sato-Tate  :  $\mu(31)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 311,\ (0:\ ),\ 0.945 + 0.324i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6465622980 + 0.1079363993i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6465622980 + 0.1079363993i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6508934136 - 0.04038357461i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6508934136 - 0.04038357461i\)

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

−25.031357541254562099758860169152, −24.29060864362532196669426945011, −23.39240526831028768271768702641, −23.09233336832274355872697158709, −22.079582246179319462188089982112, −20.32795142933281589679221617647, −19.41030322524860353537135024566, −18.84779757789648347279248879172, −17.755816542419535004485218912189, −17.20930865189350755002101601471, −16.330201601181913149444609185129, −15.07972923522333395274089052512, −14.37499082776777954831896492684, −13.5127737649512057853156774081, −12.155291293947563634917438912166, −11.29011590484007527102932736406, −10.35805954769481375475998148499, −8.93110013056767444237215093821, −7.86850826390402081468250119654, −6.93172516182684589103263896348, −6.84058891394352463923367547976, −5.0202652350282264625650357871, −4.28439311422965524578732478288, −2.15969355567265517742008743413, −0.688106684211258746566153125551, 1.04758985072310553776917781386, 2.831468511369803927079123530226, 3.92707697545113621248561544128, 4.74112101766735435462022437102, 5.85967152311878797061916097123, 7.824293632834807500798191184308, 8.62447318029160482424808581086, 9.32715045629506521803204525854, 10.72786941279368982276276262626, 11.18551027143652748743500685203, 12.153724819679846707293373705578, 12.811983865560691166998319804537, 14.5738782466582766389892908862, 15.25731472181977236495332938601, 16.52741093112550686507969362676, 17.06253077908202123937081074781, 18.076941465487972198280548472226, 19.24300003937201535164153912002, 19.871965058624032844185607365904, 20.97192475293479455411112556874, 21.4995934442471791050780404048, 22.34348169020024372252158454632, 23.25492004494127500765863091224, 24.31841770411301792420994675930, 25.467512099730890377229864870447

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