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Properties

Label 1-2011-2011.73-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.792 + 0.609i$
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  + (−0.968 − 0.247i)2-s + (−0.701 + 0.713i)3-s + (0.877 + 0.479i)4-s + (−0.812 − 0.582i)5-s + (0.855 − 0.517i)6-s + (0.573 + 0.819i)7-s + (−0.731 − 0.681i)8-s + (−0.0171 − 0.999i)9-s + (0.643 + 0.765i)10-s + (−0.962 + 0.271i)11-s + (−0.957 + 0.289i)12-s + (0.814 + 0.580i)13-s + (−0.353 − 0.935i)14-s + (0.985 − 0.171i)15-s + (0.540 + 0.841i)16-s + (−0.586 − 0.809i)17-s + ⋯
L(s)  = 1  + (−0.968 − 0.247i)2-s + (−0.701 + 0.713i)3-s + (0.877 + 0.479i)4-s + (−0.812 − 0.582i)5-s + (0.855 − 0.517i)6-s + (0.573 + 0.819i)7-s + (−0.731 − 0.681i)8-s + (−0.0171 − 0.999i)9-s + (0.643 + 0.765i)10-s + (−0.962 + 0.271i)11-s + (−0.957 + 0.289i)12-s + (0.814 + 0.580i)13-s + (−0.353 − 0.935i)14-s + (0.985 − 0.171i)15-s + (0.540 + 0.841i)16-s + (−0.586 − 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.792 + 0.609i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.792 + 0.609i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1371040602 + 0.4030652526i\)
\(L(\frac12)\) \(\approx\) \(0.1371040602 + 0.4030652526i\)
\(L(1)\) \(\approx\) \(0.4626831781 + 0.09082215838i\)
\(L(1)\) \(\approx\) \(0.4626831781 + 0.09082215838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.968 - 0.247i)T \)
3 \( 1 + (-0.701 + 0.713i)T \)
5 \( 1 + (-0.812 - 0.582i)T \)
7 \( 1 + (0.573 + 0.819i)T \)
11 \( 1 + (-0.962 + 0.271i)T \)
13 \( 1 + (0.814 + 0.580i)T \)
17 \( 1 + (-0.586 - 0.809i)T \)
19 \( 1 + (-0.977 - 0.210i)T \)
23 \( 1 + (0.989 + 0.143i)T \)
29 \( 1 + (0.0484 + 0.998i)T \)
31 \( 1 + (-0.737 + 0.674i)T \)
37 \( 1 + (0.924 - 0.380i)T \)
41 \( 1 + (-0.0983 - 0.995i)T \)
43 \( 1 + (0.107 - 0.994i)T \)
47 \( 1 + (-0.692 + 0.721i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.840 - 0.541i)T \)
61 \( 1 + (-0.248 - 0.968i)T \)
67 \( 1 + (0.996 - 0.0780i)T \)
71 \( 1 + (-0.252 + 0.967i)T \)
73 \( 1 + (-0.818 - 0.575i)T \)
79 \( 1 + (0.689 + 0.724i)T \)
83 \( 1 + (0.967 + 0.253i)T \)
89 \( 1 + (0.332 - 0.943i)T \)
97 \( 1 + (0.762 + 0.646i)T \)
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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

−19.28039766190314417713673650889, −18.610081800027068861678127779812, −18.03973208415294661916996031420, −17.46551788915314964295939535993, −16.63227322449743908314350382526, −16.12563163106391016829252598185, −15.08083544581956234261335471615, −14.74536322799609426224146015581, −13.32664088108641236032699884245, −12.96446149097179433683201620223, −11.64890054407856709050094157029, −11.14340601682606542731029517036, −10.70372798880731886324741846107, −10.13422614678783463149788204923, −8.568152075525577957114984727578, −7.96493974184855350142195685963, −7.664220769892520234988737295538, −6.619475557235350393933533588568, −6.19897910212619883238090698344, −5.12383770115817182234884079441, −4.09440829327744188399459880300, −2.87194502103274885875392781726, −1.93705495283357098946842560581, −0.88663107796827736661631573858, −0.19920846289899387040781380060, 0.724399887704885161631171009262, 1.81283646898187538932339388088, 2.89397263630012859760966872846, 3.87740051608721695389457806124, 4.81033262448706604604582797513, 5.42964832698609901445872900913, 6.561142510566185036661893880002, 7.331428710499309077344791993072, 8.34614411716191551488455598959, 8.958137107654542600540304087577, 9.35641767316410601497176275772, 10.76186565325116036164940438442, 10.95668246398887713450018286617, 11.659369335318193170493721888257, 12.4424485094327716956323821581, 12.97809863524492070051502974066, 14.588456794021117112441806091775, 15.41347677462141040887045404257, 15.819462450068224196442706873610, 16.329893623367475970443751209025, 17.194316689567795600485203228464, 17.87484535882736970836851887216, 18.54502897443541768202223386654, 19.10376203552557034515235437745, 20.31920928011155199674916951556

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