↑

Properties

Label 1-2011-2011.426-r0-0-0
Degree $1$
Conductor $2011$
Sign $-0.663 - 0.747i$
Analytic cond. $9.33904$
Root an. cond. $9.33904$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0858 − 0.996i)2-s + (−0.924 − 0.380i)3-s + (−0.985 + 0.171i)4-s + (−0.416 − 0.909i)5-s + (−0.300 + 0.953i)6-s + (0.709 − 0.704i)7-s + (0.255 + 0.966i)8-s + (0.709 + 0.704i)9-s + (−0.869 + 0.493i)10-s + (0.998 − 0.0624i)11-s + (0.976 + 0.217i)12-s + (−0.990 − 0.140i)13-s + (−0.762 − 0.646i)14-s + (0.0390 + 0.999i)15-s + (0.941 − 0.337i)16-s + (0.877 + 0.479i)17-s + ⋯
L(s)  = 1  + (−0.0858 − 0.996i)2-s + (−0.924 − 0.380i)3-s + (−0.985 + 0.171i)4-s + (−0.416 − 0.909i)5-s + (−0.300 + 0.953i)6-s + (0.709 − 0.704i)7-s + (0.255 + 0.966i)8-s + (0.709 + 0.704i)9-s + (−0.869 + 0.493i)10-s + (0.998 − 0.0624i)11-s + (0.976 + 0.217i)12-s + (−0.990 − 0.140i)13-s + (−0.762 − 0.646i)14-s + (0.0390 + 0.999i)15-s + (0.941 − 0.337i)16-s + (0.877 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.663 - 0.747i$
Analytic conductor: \(9.33904\)
Root analytic conductor: \(9.33904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (426, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (0:\ ),\ -0.663 - 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4261999644 - 0.9480874372i\)
\(L(\frac12)\) \(\approx\) \(0.4261999644 - 0.9480874372i\)
\(L(1)\) \(\approx\) \(0.5519668535 - 0.5399523799i\)
\(L(1)\) \(\approx\) \(0.5519668535 - 0.5399523799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.0858 - 0.996i)T \)
3 \( 1 + (-0.924 - 0.380i)T \)
5 \( 1 + (-0.416 - 0.909i)T \)
7 \( 1 + (0.709 - 0.704i)T \)
11 \( 1 + (0.998 - 0.0624i)T \)
13 \( 1 + (-0.990 - 0.140i)T \)
17 \( 1 + (0.877 + 0.479i)T \)
19 \( 1 + (0.458 - 0.888i)T \)
23 \( 1 + (-0.444 + 0.895i)T \)
29 \( 1 + (-0.979 - 0.201i)T \)
31 \( 1 + (-0.116 + 0.993i)T \)
37 \( 1 + (0.810 + 0.585i)T \)
41 \( 1 + (0.995 - 0.0936i)T \)
43 \( 1 + (-0.998 + 0.0468i)T \)
47 \( 1 + (0.892 - 0.451i)T \)
53 \( 1 + T \)
59 \( 1 + (0.664 - 0.747i)T \)
61 \( 1 + (0.402 + 0.915i)T \)
67 \( 1 + (0.992 + 0.124i)T \)
71 \( 1 + (0.987 - 0.155i)T \)
73 \( 1 + (0.285 + 0.958i)T \)
79 \( 1 + (0.101 + 0.994i)T \)
83 \( 1 + (0.941 + 0.337i)T \)
89 \( 1 + (-0.147 + 0.988i)T \)
97 \( 1 + (0.430 - 0.902i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.17068272172425164195812949287, −19.01182066659107263046235989425, −18.50744023714340063808967666750, −18.0255503553675547090484956031, −17.136519977390345845998916584927, −16.605979969530469527971846740750, −15.92229833657637591841947602163, −14.91909553095873539450870757276, −14.695935180305986697747822511706, −14.11895751146462091097845470543, −12.66752320336009999774380715777, −11.97456509344533523794851083020, −11.48302469816472385823346752372, −10.44823257200804515126548538057, −9.71039807507282965941396220077, −9.10830408951114160498557788553, −7.825461047454122410849111354, −7.43492264307005159977017214932, −6.46938764214105795696060163873, −5.85845599773393792909658864321, −5.134062732048714124345006396845, −4.24280170647669122142700400124, −3.6000849404677254722709626472, −2.14900914507931442855855518469, −0.75937880922935836663456583699, 0.73487205238011113367072479367, 1.24522793574264875116072062753, 2.12441238705364722216775260018, 3.654840685521012640097518450267, 4.23731948126402164532454401297, 5.10405545874214703389190109715, 5.53298736081172718141167349896, 7.026044276666047741658857644096, 7.68993684729735448348760795642, 8.419493644709735675195248886917, 9.50304486104573941330251436078, 10.034990733922290030967867438971, 11.107070163993056803772431967083, 11.56909762513100235996198270556, 12.13288904072418166197781549044, 12.79460305426714649927908305695, 13.549350351448503548913359203880, 14.29264694235651017473145675774, 15.25455376629528072229059864419, 16.55370455783570234928316068177, 16.93588421670466487684592359508, 17.48206001851777143078343059391, 18.12904333681134969925836416588, 19.17398427731610211617657323287, 19.749496432415761593081653983786

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