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Properties

Label 1-2011-2011.90-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.219 - 0.975i$
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  + (0.884 − 0.465i)2-s + (−0.673 − 0.738i)3-s + (0.566 − 0.824i)4-s + (−0.995 − 0.0905i)5-s + (−0.940 − 0.340i)6-s + (−0.659 − 0.751i)7-s + (0.116 − 0.993i)8-s + (−0.0920 + 0.995i)9-s + (−0.923 + 0.383i)10-s + (−0.0951 − 0.995i)11-s + (−0.990 + 0.137i)12-s + (−0.984 + 0.177i)13-s + (−0.933 − 0.357i)14-s + (0.604 + 0.796i)15-s + (−0.359 − 0.933i)16-s + (−0.880 + 0.474i)17-s + ⋯
L(s)  = 1  + (0.884 − 0.465i)2-s + (−0.673 − 0.738i)3-s + (0.566 − 0.824i)4-s + (−0.995 − 0.0905i)5-s + (−0.940 − 0.340i)6-s + (−0.659 − 0.751i)7-s + (0.116 − 0.993i)8-s + (−0.0920 + 0.995i)9-s + (−0.923 + 0.383i)10-s + (−0.0951 − 0.995i)11-s + (−0.990 + 0.137i)12-s + (−0.984 + 0.177i)13-s + (−0.933 − 0.357i)14-s + (0.604 + 0.796i)15-s + (−0.359 − 0.933i)16-s + (−0.880 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.219 - 0.975i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.219 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7445620706 - 0.5959493819i\)
\(L(\frac12)\) \(\approx\) \(0.7445620706 - 0.5959493819i\)
\(L(1)\) \(\approx\) \(0.6955794433 - 0.5651395648i\)
\(L(1)\) \(\approx\) \(0.6955794433 - 0.5651395648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.884 - 0.465i)T \)
3 \( 1 + (-0.673 - 0.738i)T \)
5 \( 1 + (-0.995 - 0.0905i)T \)
7 \( 1 + (-0.659 - 0.751i)T \)
11 \( 1 + (-0.0951 - 0.995i)T \)
13 \( 1 + (-0.984 + 0.177i)T \)
17 \( 1 + (-0.880 + 0.474i)T \)
19 \( 1 + (-0.842 + 0.538i)T \)
23 \( 1 + (0.980 - 0.198i)T \)
29 \( 1 + (-0.518 + 0.854i)T \)
31 \( 1 + (0.635 - 0.771i)T \)
37 \( 1 + (0.578 + 0.815i)T \)
41 \( 1 + (0.576 + 0.817i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (-0.970 + 0.241i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.480 - 0.876i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.905 + 0.423i)T \)
71 \( 1 + (0.996 - 0.0811i)T \)
73 \( 1 + (0.754 - 0.656i)T \)
79 \( 1 + (-0.871 - 0.490i)T \)
83 \( 1 + (-0.998 - 0.0531i)T \)
89 \( 1 + (0.626 + 0.779i)T \)
97 \( 1 + (0.698 + 0.715i)T \)
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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

−19.84564987220611476431486082058, −19.55868936531822059889696303430, −18.260853235260720830193709044478, −17.43857313995000909040458420401, −16.86987644623572417340599307963, −15.917096402611826243584655415638, −15.50595181018471955677670426313, −15.086236771246079590954718609950, −14.42825442156071330888133896035, −12.94334106187766080448749555484, −12.62612913740708598742992982643, −11.85804011569481972683809391848, −11.2971985889603160158048466469, −10.469426670067110518523464011077, −9.38398796288764099364494707204, −8.752764432125612429347742131986, −7.53397214134694902587650148731, −6.92588451473671950027663188602, −6.23066373103758775371435582987, −5.18938942587291384317770698166, −4.640242157830478697713376318083, −4.03467348600967799238192612237, −2.987504256766695735463390726588, −2.34728176994929456096009260094, −0.26854172890883349875573485143, 0.46896850660766825912193312316, 1.32741042298873997522888720571, 2.56247462871742915990339732641, 3.32746498863312420142144171205, 4.35677366893711084335720980532, 4.81241824232906460999651527319, 6.05976882470428066894227770845, 6.52310482075229903216722703839, 7.32940848637609206796182824695, 8.0398745504648902137120584128, 9.24020737444940743522596038777, 10.41604913893061454723648589874, 10.99081945247191219248198741170, 11.44683108223222881490391862193, 12.44376209828742089361050421697, 12.82729488963225756105045848161, 13.45492708527969282738789364557, 14.34869429937217793525514799706, 15.11847019794545201832995130589, 16.00451763238309580849233305572, 16.68643439981209491858047689956, 17.14061995365444301503148034484, 18.59512471755197612189232203428, 19.03471805224590033758912243007, 19.61920507215341371256342439340

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