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Properties

Label 1-2011-2011.1229-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.975 + 0.218i$
Analytic cond. $216.111$
Root an. cond. $216.111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.975 + 0.218i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.975 + 0.218i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4977664621 - 4.503558137i\)
\(L(\frac12)\) \(\approx\) \(-0.4977664621 - 4.503558137i\)
\(L(1)\) \(\approx\) \(1.573660286 - 1.525473525i\)
\(L(1)\) \(\approx\) \(1.573660286 - 1.525473525i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.782 - 0.622i)T \)
3 \( 1 + (0.716 - 0.697i)T \)
5 \( 1 + (0.508 + 0.861i)T \)
7 \( 1 + (0.942 - 0.334i)T \)
11 \( 1 + (-0.910 - 0.412i)T \)
13 \( 1 + (0.576 - 0.817i)T \)
17 \( 1 + (-0.631 + 0.775i)T \)
19 \( 1 + (0.863 + 0.504i)T \)
23 \( 1 + (-0.805 - 0.592i)T \)
29 \( 1 + (0.876 + 0.482i)T \)
31 \( 1 + (0.144 - 0.989i)T \)
37 \( 1 + (0.444 - 0.895i)T \)
41 \( 1 + (-0.999 - 0.00937i)T \)
43 \( 1 + (-0.952 + 0.304i)T \)
47 \( 1 + (-0.780 - 0.625i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.855 + 0.517i)T \)
61 \( 1 + (-0.947 + 0.319i)T \)
67 \( 1 + (-0.918 - 0.395i)T \)
71 \( 1 + (-0.680 + 0.732i)T \)
73 \( 1 + (-0.230 - 0.973i)T \)
79 \( 1 + (-0.248 - 0.968i)T \)
83 \( 1 + (0.984 + 0.174i)T \)
89 \( 1 + (0.928 - 0.372i)T \)
97 \( 1 + (0.869 - 0.493i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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