Properties

Degree 1
Conductor 2011
Sign $-0.975 - 0.218i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.782 + 0.622i)2-s + (0.716 + 0.697i)3-s + (0.224 + 0.974i)4-s + (0.508 − 0.861i)5-s + (0.126 + 0.991i)6-s + (0.942 + 0.334i)7-s + (−0.430 + 0.902i)8-s + (0.0265 + 0.999i)9-s + (0.933 − 0.357i)10-s + (−0.910 + 0.412i)11-s + (−0.518 + 0.854i)12-s + (0.576 + 0.817i)13-s + (0.529 + 0.848i)14-s + (0.964 − 0.262i)15-s + (−0.899 + 0.437i)16-s + (−0.631 − 0.775i)17-s + ⋯
L(s,χ)  = 1  + (0.782 + 0.622i)2-s + (0.716 + 0.697i)3-s + (0.224 + 0.974i)4-s + (0.508 − 0.861i)5-s + (0.126 + 0.991i)6-s + (0.942 + 0.334i)7-s + (−0.430 + 0.902i)8-s + (0.0265 + 0.999i)9-s + (0.933 − 0.357i)10-s + (−0.910 + 0.412i)11-s + (−0.518 + 0.854i)12-s + (0.576 + 0.817i)13-s + (0.529 + 0.848i)14-s + (0.964 − 0.262i)15-s + (−0.899 + 0.437i)16-s + (−0.631 − 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.975 - 0.218i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.975 - 0.218i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $-0.975 - 0.218i$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (18, \cdot )$
Sato-Tate  :  $\mu(2010)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2011,\ (1:\ ),\ -0.975 - 0.218i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.4977664621 + 4.503558137i$
$L(\frac12,\chi)$  $\approx$  $-0.4977664621 + 4.503558137i$
$L(\chi,1)$  $\approx$  1.573660286 + 1.525473525i
$L(1,\chi)$  $\approx$  1.573660286 + 1.525473525i

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

−19.56905613316517212824281534255, −18.45415074198400041006693458464, −18.274427172518938427335720708557, −17.68371173280901654273154508810, −16.25802314092447175457674446094, −15.13873016277425037166920658023, −14.903570791881352301582846551144, −13.99464699462348981571565314051, −13.53985723621424318834721178660, −13.041586094351714079016326733144, −12.01842708021736933561312202649, −11.29038087302163244457692813384, −10.42056715336600055042397864185, −10.120637549527165913249331703985, −8.797244958703526568657241523100, −8.00354643717171092855825612319, −7.28990224485848019515353481843, −6.22510238532518216419890906769, −5.79143612833098048812187261553, −4.68436461208294909031706933244, −3.592972917659357913732964473526, −3.03138184354695841100164745428, −2.12873221284561809243875600396, −1.5491407246410311720136534430, −0.43502485248098532279236879897, 1.55357903963064198953540921575, 2.34328085850030660175561275276, 3.18830014644552639595774254446, 4.39838985077787918205967392948, 4.831700997581742204144127990129, 5.29897460175375390541276444361, 6.37673123804493989082882803579, 7.5038218927212871883579236742, 8.174539472159413368771551024222, 8.78771155598274458920014642628, 9.46232548348732761905683352515, 10.453312288070718865135858891082, 11.557862448793796701479076011557, 12.01760711205710806981835009035, 13.26782470745029284397668840747, 13.673101657930709814352500999177, 14.13151080265679853803242399629, 15.149137010628822937909717872325, 15.77462547238291476615249217743, 16.125857697044163260667175744997, 17.058306247276944136821977782727, 17.86770994462598161247728883299, 18.4110759098212655844745712315, 19.93857385012767266800794445455, 20.37544726818625275660614198287

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