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Properties

Label 1-4009-4009.1093-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.944 + 0.328i$
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 + (−0.969 − 0.246i)5-s + (−0.318 − 0.947i)6-s + (−0.0747 + 0.997i)7-s + (0.955 + 0.294i)8-s + (0.980 + 0.198i)9-s + (0.173 + 0.984i)10-s + (0.826 − 0.563i)11-s + (−0.733 + 0.680i)12-s + (0.998 + 0.0498i)13-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)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 + (−0.969 − 0.246i)5-s + (−0.318 − 0.947i)6-s + (−0.0747 + 0.997i)7-s + (0.955 + 0.294i)8-s + (0.980 + 0.198i)9-s + (0.173 + 0.984i)10-s + (0.826 − 0.563i)11-s + (−0.733 + 0.680i)12-s + (0.998 + 0.0498i)13-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)15-s + (−0.124 − 0.992i)16-s + (−0.456 + 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.944 + 0.328i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1093, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.726381277 + 0.2919021121i\)
\(L(\frac12)\) \(\approx\) \(1.726381277 + 0.2919021121i\)
\(L(1)\) \(\approx\) \(1.093682357 - 0.1635599185i\)
\(L(1)\) \(\approx\) \(1.093682357 - 0.1635599185i\)

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 \( 1 + (-0.969 - 0.246i)T \)
7 \( 1 + (-0.0747 + 0.997i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (0.998 + 0.0498i)T \)
17 \( 1 + (-0.456 + 0.889i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.456 + 0.889i)T \)
31 \( 1 + (0.955 + 0.294i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (0.456 - 0.889i)T \)
43 \( 1 + (0.542 + 0.840i)T \)
47 \( 1 + (0.878 + 0.478i)T \)
53 \( 1 + (0.853 - 0.521i)T \)
59 \( 1 + (-0.456 + 0.889i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.995 - 0.0995i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.0249 + 0.999i)T \)
79 \( 1 + (-0.921 - 0.388i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.270 - 0.962i)T \)
97 \( 1 + (-0.853 - 0.521i)T \)
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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

−18.48267839191098937070417539833, −17.80178544159882226065980088544, −17.02579941502769961992776038335, −16.187380156459694363850651917165, −15.70782678330248008665077272228, −15.16500350288278539580289837905, −14.39369575475133944724344599356, −13.81986806000785275452004150061, −13.427370157248639709159715452, −12.36638366556892565265386473764, −11.530956790065756375665425879918, −10.561323896919858255185488183775, −10.024892525345558878736305900193, −9.22674395407145598330842301320, −8.474581849043170421519053689548, −8.02131287748410469737487039056, −7.192914207625934525485782021223, −6.863709420018946586724930854400, −6.11400836659475005891283146443, −4.5871525957483321549924757678, −4.243909155066565228032125730075, −3.65203929978097085194984673816, −2.54977886360684493390883373118, −1.35282938911658223393506445485, −0.60873573460412867304542871000, 1.05342125948136155629296599513, 1.72626324139357926483089168492, 2.67252818806209468803805859441, 3.457572052431308318985853744500, 3.888345662076711513134296252011, 4.57867090710915889932151372490, 5.73737026309592823294138101599, 6.77485923020478021107903543844, 7.72109826894665130118260007410, 8.47567617837896847102972565320, 8.7307560665580573686101244855, 9.234129331240713430384587592373, 10.24035327441535080297981719472, 10.99663559949008056619915881751, 11.61954641324656103581242560026, 12.40715425787832033641094656643, 12.76097321111566910421617615032, 13.76911232475584599559286858915, 14.240197475287339746127962502014, 15.20152930406298137086130975716, 15.87148097802493690156117229811, 16.24370396246648447110282125986, 17.366226561016120707906146952509, 18.09647306767594200409788496890, 18.89165206093774292946953939560

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