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Properties

Label 1-2011-2011.82-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.266 - 0.963i$
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  + (−0.224 − 0.974i)2-s + (0.942 − 0.334i)3-s + (−0.899 + 0.437i)4-s + (−0.981 + 0.189i)5-s + (−0.537 − 0.843i)6-s + (0.998 + 0.0531i)7-s + (0.628 + 0.777i)8-s + (0.776 − 0.629i)9-s + (0.405 + 0.914i)10-s + (0.0920 − 0.995i)11-s + (−0.701 + 0.713i)12-s + (0.825 − 0.564i)13-s + (−0.172 − 0.984i)14-s + (−0.862 + 0.506i)15-s + (0.616 − 0.787i)16-s + (0.411 + 0.911i)17-s + ⋯
L(s)  = 1  + (−0.224 − 0.974i)2-s + (0.942 − 0.334i)3-s + (−0.899 + 0.437i)4-s + (−0.981 + 0.189i)5-s + (−0.537 − 0.843i)6-s + (0.998 + 0.0531i)7-s + (0.628 + 0.777i)8-s + (0.776 − 0.629i)9-s + (0.405 + 0.914i)10-s + (0.0920 − 0.995i)11-s + (−0.701 + 0.713i)12-s + (0.825 − 0.564i)13-s + (−0.172 − 0.984i)14-s + (−0.862 + 0.506i)15-s + (0.616 − 0.787i)16-s + (0.411 + 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.266 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.071951301 - 2.722692824i\)
\(L(\frac12)\) \(\approx\) \(2.071951301 - 2.722692824i\)
\(L(1)\) \(\approx\) \(1.178571776 - 0.8220122432i\)
\(L(1)\) \(\approx\) \(1.178571776 - 0.8220122432i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.224 - 0.974i)T \)
3 \( 1 + (0.942 - 0.334i)T \)
5 \( 1 + (-0.981 + 0.189i)T \)
7 \( 1 + (0.998 + 0.0531i)T \)
11 \( 1 + (0.0920 - 0.995i)T \)
13 \( 1 + (0.825 - 0.564i)T \)
17 \( 1 + (0.411 + 0.911i)T \)
19 \( 1 + (0.909 - 0.415i)T \)
23 \( 1 + (0.999 - 0.0125i)T \)
29 \( 1 + (-0.0640 - 0.997i)T \)
31 \( 1 + (0.943 + 0.331i)T \)
37 \( 1 + (0.604 - 0.796i)T \)
41 \( 1 + (0.326 + 0.945i)T \)
43 \( 1 + (-0.999 + 0.00937i)T \)
47 \( 1 + (-0.396 - 0.917i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.896 + 0.443i)T \)
61 \( 1 + (0.999 + 0.0218i)T \)
67 \( 1 + (-0.687 + 0.726i)T \)
71 \( 1 + (0.925 + 0.377i)T \)
73 \( 1 + (-0.151 + 0.988i)T \)
79 \( 1 + (0.729 - 0.684i)T \)
83 \( 1 + (-0.558 + 0.829i)T \)
89 \( 1 + (-0.433 + 0.901i)T \)
97 \( 1 + (-0.513 - 0.858i)T \)
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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

−20.008110703386016564186423097958, −19.075762801150171060376124180806, −18.52921465023706337075014445287, −17.87724589234923644765618702371, −16.81984860841140032956758017042, −16.16201988520327362226816220429, −15.57176044298260242276141522142, −14.89301923603964562203484893532, −14.386977555953688830909585881473, −13.7059447082551288240584263467, −12.84249772449607809928490069185, −11.83107490162034233812932999342, −11.01955844670501745537869399637, −10.00059398543340478349255091461, −9.272309089426689687639127840623, −8.578380177044189254996560851108, −7.930142576287144249597453100946, −7.37389366785880563242349653975, −6.71983796550361818389236442816, −5.118666681794408807514766975888, −4.790862259725388811744547143142, −3.97521258310253517881395491958, −3.15646616026327175056667776423, −1.67412420065498680427610468096, −0.9116653644125197422209506497, 0.860359943341489920009975873290, 1.1346995039886085893547505900, 2.45936799322131481127711777809, 3.211130641162085527974504747016, 3.78869670815365508388813039603, 4.59269110517842184437344415698, 5.69467816264515044522116022930, 7.028865627116492436080456229116, 8.02217498646368994521888910752, 8.23834306331233383553847586191, 8.84929687057024754602169882159, 9.91077207497038452095681038571, 10.827388834879195986943730299961, 11.39865047542716619346359504381, 11.99699737827711237654009065312, 12.99050148313128748329338008847, 13.52090944103293347405272966432, 14.34379529138016871106508576977, 14.9720006041620877573922274615, 15.74765001513864669507604054311, 16.75708439506277577080084015268, 17.749568149445769439686794446, 18.38679774585302877118216939558, 18.945609581570264545199739772411, 19.58788695953141816945366145797

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