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Properties

Label 1-4011-4011.194-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.984 + 0.176i$
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  + (0.982 + 0.186i)2-s + (0.930 + 0.366i)4-s + (0.350 + 0.936i)5-s + (0.846 + 0.533i)8-s + (0.170 + 0.985i)10-s + (−0.137 − 0.990i)11-s + (0.724 − 0.689i)13-s + (0.731 + 0.681i)16-s + (0.509 + 0.860i)17-s + (0.170 − 0.985i)19-s + (−0.0165 + 0.999i)20-s + (0.0495 − 0.998i)22-s + (0.989 − 0.142i)23-s + (−0.754 + 0.656i)25-s + (0.840 − 0.542i)26-s + ⋯
L(s)  = 1  + (0.982 + 0.186i)2-s + (0.930 + 0.366i)4-s + (0.350 + 0.936i)5-s + (0.846 + 0.533i)8-s + (0.170 + 0.985i)10-s + (−0.137 − 0.990i)11-s + (0.724 − 0.689i)13-s + (0.731 + 0.681i)16-s + (0.509 + 0.860i)17-s + (0.170 − 0.985i)19-s + (−0.0165 + 0.999i)20-s + (0.0495 − 0.998i)22-s + (0.989 − 0.142i)23-s + (−0.754 + 0.656i)25-s + (0.840 − 0.542i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.984 + 0.176i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.984 + 0.176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.221067676 + 0.3753601876i\)
\(L(\frac12)\) \(\approx\) \(4.221067676 + 0.3753601876i\)
\(L(1)\) \(\approx\) \(2.233835248 + 0.3311782684i\)
\(L(1)\) \(\approx\) \(2.233835248 + 0.3311782684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.982 + 0.186i)T \)
5 \( 1 + (0.350 + 0.936i)T \)
11 \( 1 + (-0.137 - 0.990i)T \)
13 \( 1 + (0.724 - 0.689i)T \)
17 \( 1 + (0.509 + 0.860i)T \)
19 \( 1 + (0.170 - 0.985i)T \)
23 \( 1 + (0.989 - 0.142i)T \)
29 \( 1 + (-0.601 - 0.799i)T \)
31 \( 1 + (-0.451 - 0.892i)T \)
37 \( 1 + (0.451 - 0.892i)T \)
41 \( 1 + (-0.0825 - 0.996i)T \)
43 \( 1 + (0.746 - 0.665i)T \)
47 \( 1 + (0.0715 - 0.997i)T \)
53 \( 1 + (-0.471 + 0.882i)T \)
59 \( 1 + (0.159 - 0.987i)T \)
61 \( 1 + (0.917 - 0.396i)T \)
67 \( 1 + (0.815 - 0.578i)T \)
71 \( 1 + (-0.922 + 0.386i)T \)
73 \( 1 + (0.644 + 0.764i)T \)
79 \( 1 + (-0.992 - 0.120i)T \)
83 \( 1 + (0.371 + 0.928i)T \)
89 \( 1 + (0.411 + 0.911i)T \)
97 \( 1 + (-0.965 + 0.261i)T \)
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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

−18.51863225893643423001720995053, −17.769709295100932325556532479771, −16.81482070982561476568718193073, −16.2746396153925727270356211938, −15.87793555630198941469805820564, −14.7915778878403787184088488062, −14.39482910356117084926099089804, −13.5678538410992491826546753556, −12.94546368147290437756743275598, −12.47376585870425472007488299405, −11.74094609128444279763363332962, −11.13393079968380275127888939277, −10.1400773717078051228288487114, −9.58936413069239939401032985612, −8.83395290941543982008263039664, −7.8116449477639106585695601430, −7.13325974364745147785291293750, −6.322803155274038257136673744843, −5.54056909530252437276790583147, −4.91014654573879816109765712898, −4.36127907769043963977433233041, −3.470059737688278289611657506593, −2.63976583413724544335548922988, −1.520878891068018378324243744553, −1.27675655345941825817134100080, 0.85712213345779361060121545389, 2.106995519494629658344982800829, 2.731604340047940060885615167026, 3.564256245816694495348904309625, 3.951954775358395080917877201311, 5.344342600867412629919549609295, 5.689323804881220239024167498774, 6.37522154870682720752895797660, 7.12995678681995211182829746085, 7.82160076736994644420564356907, 8.59036344990594529135833820700, 9.58849869742850172533390682770, 10.64294075688804490227598100664, 10.967473074096357197227424916613, 11.47671575127818582838022794128, 12.58924049525571711610067154183, 13.175533173445447747108849454930, 13.7209714348511945426705774233, 14.34482805537101029889042382526, 15.15360156523135456984652984457, 15.46514564662669270282777706080, 16.31827333356692059344385306968, 17.15086199816944651491764558148, 17.59405392635756145435895545538, 18.77681340918310369906322162357

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