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Properties

Label 1-4011-4011.395-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.0892 - 0.996i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0892 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0892 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.0892 - 0.996i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.0892 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8319709470 - 0.9098953582i\)
\(L(\frac12)\) \(\approx\) \(0.8319709470 - 0.9098953582i\)
\(L(1)\) \(\approx\) \(0.7796240294 - 0.5152312665i\)
\(L(1)\) \(\approx\) \(0.7796240294 - 0.5152312665i\)

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

−18.5322277362760668871122305637, −17.74380212325440489747457301190, −17.37857387102616025743999148796, −16.644793874556092850326928562094, −15.766298872828796798143429546490, −15.35315938476386295795129330861, −14.4364702454083425831314925294, −14.20097188003051066593657475190, −13.27940281257835157164896060102, −12.89511346661050499097062858828, −11.73754097314577180502652734838, −11.03657021898266839145696005809, −10.3406029203025453007705868264, −9.37449532145787044353190831956, −9.026897279407320598740256450191, −7.970825478791356376225690600912, −7.383361878574302160916985035348, −6.6499144654567761434854356483, −6.14909076522228981622983373680, −5.33179210971477087489991222277, −4.59662791113162519303675916338, −3.69781696448402264238420792497, −2.880653670231637898585505384690, −2.13084168801768992016884534459, −0.585792344577637313624069340412, 0.55345491550729933267435242939, 1.68956894576343237340716517969, 2.17138424956307396507929733375, 2.996872030233567954424423451752, 4.27914298803803563402389448716, 4.564570387512180277687565452596, 5.214993577100809115411946271, 6.17334132383728684236586926358, 7.05110809093834078354275045196, 8.221630221223792593117750383638, 8.6342049358626762649259150470, 9.42472758193746664009348139195, 10.14631699923030503284957583629, 10.50308242824049861217763079748, 11.58235467071403908687549801849, 12.33179380264910560175430495390, 12.64699924104229711439169630946, 13.25670362368532284011424971781, 14.04484778540869481821850024153, 14.86330838162469054104466278929, 15.33132701979779406594317059513, 16.53673130198854835592818947283, 17.10001661473142120730124956330, 17.69114243752777637904561752479, 18.23896369370584589951987515267

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