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Properties

Label 1-4011-4011.338-r1-0-0
Degree $1$
Conductor $4011$
Sign $-0.719 + 0.694i$
Analytic cond. $431.041$
Root an. cond. $431.041$
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  + (0.126 + 0.991i)2-s + (−0.968 + 0.250i)4-s + (−0.350 − 0.936i)5-s + (−0.371 − 0.928i)8-s + (0.884 − 0.466i)10-s + (−0.137 − 0.990i)11-s + (0.991 − 0.131i)13-s + (0.874 − 0.485i)16-s + (0.917 + 0.396i)17-s + (0.884 + 0.466i)19-s + (0.574 + 0.818i)20-s + (0.965 − 0.261i)22-s + (0.441 + 0.897i)23-s + (−0.754 + 0.656i)25-s + (0.256 + 0.966i)26-s + ⋯
L(s)  = 1  + (0.126 + 0.991i)2-s + (−0.968 + 0.250i)4-s + (−0.350 − 0.936i)5-s + (−0.371 − 0.928i)8-s + (0.884 − 0.466i)10-s + (−0.137 − 0.990i)11-s + (0.991 − 0.131i)13-s + (0.874 − 0.485i)16-s + (0.917 + 0.396i)17-s + (0.884 + 0.466i)19-s + (0.574 + 0.818i)20-s + (0.965 − 0.261i)22-s + (0.441 + 0.897i)23-s + (−0.754 + 0.656i)25-s + (0.256 + 0.966i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.719 + 0.694i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ -0.719 + 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6554378270 + 1.622932825i\)
\(L(\frac12)\) \(\approx\) \(0.6554378270 + 1.622932825i\)
\(L(1)\) \(\approx\) \(0.9589047097 + 0.4157154296i\)
\(L(1)\) \(\approx\) \(0.9589047097 + 0.4157154296i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.126 + 0.991i)T \)
5 \( 1 + (-0.350 - 0.936i)T \)
11 \( 1 + (-0.137 - 0.990i)T \)
13 \( 1 + (0.991 - 0.131i)T \)
17 \( 1 + (0.917 + 0.396i)T \)
19 \( 1 + (0.884 + 0.466i)T \)
23 \( 1 + (0.441 + 0.897i)T \)
29 \( 1 + (0.956 + 0.293i)T \)
31 \( 1 + (0.451 + 0.892i)T \)
37 \( 1 + (0.451 - 0.892i)T \)
41 \( 1 + (0.0825 + 0.996i)T \)
43 \( 1 + (-0.213 + 0.976i)T \)
47 \( 1 + (-0.528 - 0.849i)T \)
53 \( 1 + (-0.984 - 0.175i)T \)
59 \( 1 + (-0.988 + 0.153i)T \)
61 \( 1 + (0.0935 + 0.995i)T \)
67 \( 1 + (-0.319 + 0.947i)T \)
71 \( 1 + (0.518 - 0.854i)T \)
73 \( 1 + (0.970 + 0.240i)T \)
79 \( 1 + (0.731 + 0.681i)T \)
83 \( 1 + (-0.997 + 0.0660i)T \)
89 \( 1 + (0.868 + 0.495i)T \)
97 \( 1 + (-0.934 - 0.355i)T \)
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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

−18.33377587417793300629861576206, −17.644830607136007224542851062881, −16.89957318058229694273377499875, −15.74968326937622098106920770286, −15.31717556674207565616032766751, −14.41234066921270655830950088969, −13.955934420431568569435561803882, −13.26756525293332934877060035657, −12.32203674165165816720676476543, −11.90107827925986771333888310982, −11.11553472699180075055892441636, −10.60000602213238916813067150375, −9.83839048513620143679813643632, −9.35060340787077757402012336316, −8.26317084989389536039325789739, −7.70795552790094560202078076026, −6.752214938000287464432101122845, −6.04019927645591731165548270925, −4.99763363249735269377922059873, −4.39227101506326531218749049856, −3.48313381366079849305181581723, −2.92883802734570910947046631594, −2.18265696378808751790223606858, −1.18262733459503910714881385616, −0.31189210203147331995963439744, 0.99493459222919978578636884277, 1.213787447471493798528771003468, 3.207838294526064599600762308830, 3.510051036373338041699759515083, 4.50250343801807601212665658588, 5.2793873910565574294038381956, 5.78591347834787957810071322749, 6.48790230186028427928933709244, 7.56118982359492649060722621125, 8.11596005925653901057371384496, 8.57368271322583185621775419031, 9.33628358224036437884759330947, 10.041524640160114723754321442595, 11.08021500461543957056991599674, 11.856378583490689209942481464392, 12.59734702874641251267294211825, 13.23043200335717769448723292168, 13.845725130796038699850908354151, 14.43200486976535234732214495364, 15.42911933604533477306621985354, 15.92047473699644467607076427739, 16.49393607386496973106094773741, 16.85257792953247538551971475419, 17.95615473777771954760957958355, 18.23481692541076997591883934229

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