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Properties

Label 1-4009-4009.101-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.392 - 0.919i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
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  + (−0.411 − 0.911i)2-s + (0.995 + 0.0995i)3-s + (−0.661 + 0.749i)4-s + (0.698 − 0.715i)5-s + (−0.318 − 0.947i)6-s + (−0.900 + 0.433i)7-s + (0.955 + 0.294i)8-s + (0.980 + 0.198i)9-s + (−0.939 − 0.342i)10-s + (0.826 − 0.563i)11-s + (−0.733 + 0.680i)12-s + (0.456 + 0.889i)13-s + (0.766 + 0.642i)14-s + (0.766 − 0.642i)15-s + (−0.124 − 0.992i)16-s + (0.456 − 0.889i)17-s + ⋯
L(s)  = 1  + (−0.411 − 0.911i)2-s + (0.995 + 0.0995i)3-s + (−0.661 + 0.749i)4-s + (0.698 − 0.715i)5-s + (−0.318 − 0.947i)6-s + (−0.900 + 0.433i)7-s + (0.955 + 0.294i)8-s + (0.980 + 0.198i)9-s + (−0.939 − 0.342i)10-s + (0.826 − 0.563i)11-s + (−0.733 + 0.680i)12-s + (0.456 + 0.889i)13-s + (0.766 + 0.642i)14-s + (0.766 − 0.642i)15-s + (−0.124 − 0.992i)16-s + (0.456 − 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.392 - 0.919i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.392 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182975789 - 1.790721211i\)
\(L(\frac12)\) \(\approx\) \(1.182975789 - 1.790721211i\)
\(L(1)\) \(\approx\) \(1.120347245 - 0.6496901718i\)
\(L(1)\) \(\approx\) \(1.120347245 - 0.6496901718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.411 - 0.911i)T \)
3 \( 1 + (0.995 + 0.0995i)T \)
5 \( 1 + (0.698 - 0.715i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (0.456 + 0.889i)T \)
17 \( 1 + (0.456 - 0.889i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.456 + 0.889i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (-0.998 + 0.0498i)T \)
43 \( 1 + (0.456 - 0.889i)T \)
47 \( 1 + (-0.853 + 0.521i)T \)
53 \( 1 + (-0.0249 - 0.999i)T \)
59 \( 1 + (0.456 - 0.889i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.583 - 0.811i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.878 - 0.478i)T \)
79 \( 1 + (-0.797 + 0.603i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.969 + 0.246i)T \)
97 \( 1 + (0.878 - 0.478i)T \)
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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

−18.66602265279555057673001266404, −17.93890710730180532437283316418, −17.41654041463998898168176167626, −16.6881885039598037025247103880, −15.80332467103783524692383264438, −15.2838802848913768283698489841, −14.603614736315729007723286584624, −14.126561014082068550611727152052, −13.35707643770842528009307836014, −13.031367034838606386934948903720, −11.981515767659619105533685091233, −10.50832409420946547364198338310, −10.19732520270908447996814786004, −9.66365467632818154932724319578, −8.977990579726533434858500217604, −8.12511407653755536802157169456, −7.559922400592366858456065438144, −6.64980303645705307610509124581, −6.40068131972092916585963228635, −5.55017729509078601975491838556, −4.34263235256099639172676943199, −3.63340278582929816563445847782, −2.94811479269938783198227792078, −1.777222979430634718653302571211, −1.150505252402194610546191262373, 0.632617255111420701767575129938, 1.63411996127996617850012471648, 2.18836121536536473638635619215, 3.06673152008798735249818482870, 3.7389411131530512403458320293, 4.3989838785568954237724814754, 5.38355011533809882817740447381, 6.36970300263844070509053803345, 7.16272576125009480544408760029, 8.256852205771017367777363554115, 8.77904485196265152313893378767, 9.31460666940410553372366293157, 9.70499777801743909901844278433, 10.40681448962679506796003454905, 11.49690095353583305451580774879, 12.15668863956942295377688496903, 12.785734966862426559583723145538, 13.465499237614182542124816023575, 13.99805259358032085470388856205, 14.48063988976830473566629816174, 15.83775116421103638523900851782, 16.40205773508836515720363564010, 16.70917643683124724077089364321, 17.89045195614389298570929111876, 18.45069520226204866609442032517

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