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Properties

Label 1-2011-2011.91-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.846 + 0.533i$
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  + (−0.992 + 0.124i)2-s + (0.652 − 0.757i)3-s + (0.968 − 0.247i)4-s + (0.810 − 0.585i)5-s + (−0.553 + 0.833i)6-s + (0.147 − 0.988i)7-s + (−0.930 + 0.366i)8-s + (−0.147 − 0.988i)9-s + (−0.731 + 0.681i)10-s + (0.720 + 0.693i)11-s + (0.444 − 0.895i)12-s + (0.591 − 0.806i)13-s + (−0.0234 + 0.999i)14-s + (0.0858 − 0.996i)15-s + (0.877 − 0.479i)16-s + (−0.987 − 0.155i)17-s + ⋯
L(s)  = 1  + (−0.992 + 0.124i)2-s + (0.652 − 0.757i)3-s + (0.968 − 0.247i)4-s + (0.810 − 0.585i)5-s + (−0.553 + 0.833i)6-s + (0.147 − 0.988i)7-s + (−0.930 + 0.366i)8-s + (−0.147 − 0.988i)9-s + (−0.731 + 0.681i)10-s + (0.720 + 0.693i)11-s + (0.444 − 0.895i)12-s + (0.591 − 0.806i)13-s + (−0.0234 + 0.999i)14-s + (0.0858 − 0.996i)15-s + (0.877 − 0.479i)16-s + (−0.987 − 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.846 + 0.533i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.846 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3573407002 - 1.237416121i\)
\(L(\frac12)\) \(\approx\) \(-0.3573407002 - 1.237416121i\)
\(L(1)\) \(\approx\) \(0.8140792767 - 0.5233731281i\)
\(L(1)\) \(\approx\) \(0.8140792767 - 0.5233731281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.992 + 0.124i)T \)
3 \( 1 + (0.652 - 0.757i)T \)
5 \( 1 + (0.810 - 0.585i)T \)
7 \( 1 + (0.147 - 0.988i)T \)
11 \( 1 + (0.720 + 0.693i)T \)
13 \( 1 + (0.591 - 0.806i)T \)
17 \( 1 + (-0.987 - 0.155i)T \)
19 \( 1 + (0.742 - 0.670i)T \)
23 \( 1 + (-0.239 + 0.970i)T \)
29 \( 1 + (-0.132 - 0.991i)T \)
31 \( 1 + (0.255 + 0.966i)T \)
37 \( 1 + (-0.194 + 0.980i)T \)
41 \( 1 + (-0.912 + 0.409i)T \)
43 \( 1 + (-0.209 - 0.977i)T \)
47 \( 1 + (-0.513 - 0.858i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.792 + 0.610i)T \)
61 \( 1 + (-0.999 + 0.0312i)T \)
67 \( 1 + (-0.0390 + 0.999i)T \)
71 \( 1 + (0.941 + 0.337i)T \)
73 \( 1 + (-0.00781 + 0.999i)T \)
79 \( 1 + (-0.752 + 0.658i)T \)
83 \( 1 + (0.877 + 0.479i)T \)
89 \( 1 + (-0.801 + 0.597i)T \)
97 \( 1 + (-0.344 - 0.938i)T \)
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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

−20.17510081672793768537602315479, −19.33034824057074634443015702741, −18.64262210897708942585174347510, −18.27331403682155437415934473254, −17.29377613072644068001390432143, −16.49038373760085014716416455093, −15.980507065956158251894490302, −15.14045269198836269284339326025, −14.436772378467765975554743917395, −13.894178156894095292722866670479, −12.795661171219217212343852795764, −11.70411455257297803778219335722, −11.05592450685954395852118724856, −10.506873091881619012509784463020, −9.3668600915514649891758158091, −9.24442664220507742478592506757, −8.510978508584934979778415065431, −7.677291804968885973501309935886, −6.36921120049509502585772414302, −6.159976912957185329328344597383, −4.92547493003805726601352573777, −3.65481320636447097109127695345, −2.99315361955728146989208592413, −2.08006825532500284599148606356, −1.5503639271761348408703497481, 0.2495815801898764145164631804, 1.27156536348442822933141030519, 1.57249530180250895965683856027, 2.66468653069948032931324692310, 3.61851864013704881574808822321, 4.85044665080589809594196394118, 5.94646170673622171225932867462, 6.782670343823631713837531870891, 7.21853666503055251712299621172, 8.2226884008511492574371380055, 8.70855033782336305456777888091, 9.66662454176426910721804846380, 9.97218297779062777439248689740, 11.13137562328161431966421528116, 11.87494752056748307435063109196, 12.77662196133858849625108008297, 13.66944963372168188186044079275, 13.899354171201248936745634482426, 15.16012609053696249589846053582, 15.613651149057977842135515932399, 16.78248022668361510615992445424, 17.42407890754140551271796553784, 17.76995431172673924313565758944, 18.38190734606321261383098697864, 19.530681963442209232308637912935

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