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Properties

Label 1-2009-2009.6-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.482 + 0.875i$
Analytic cond. $9.32975$
Root an. cond. $9.32975$
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  + (−0.657 + 0.753i)2-s + (0.993 + 0.111i)3-s + (−0.134 − 0.990i)4-s + (0.351 + 0.936i)5-s + (−0.738 + 0.674i)6-s + (0.834 + 0.550i)8-s + (0.974 + 0.222i)9-s + (−0.936 − 0.351i)10-s + (0.244 − 0.969i)11-s + (−0.0224 − 0.999i)12-s + (0.997 − 0.0672i)13-s + (0.244 + 0.969i)15-s + (−0.963 + 0.266i)16-s + (0.795 − 0.605i)17-s + (−0.809 + 0.587i)18-s + (−0.156 − 0.987i)19-s + ⋯
L(s)  = 1  + (−0.657 + 0.753i)2-s + (0.993 + 0.111i)3-s + (−0.134 − 0.990i)4-s + (0.351 + 0.936i)5-s + (−0.738 + 0.674i)6-s + (0.834 + 0.550i)8-s + (0.974 + 0.222i)9-s + (−0.936 − 0.351i)10-s + (0.244 − 0.969i)11-s + (−0.0224 − 0.999i)12-s + (0.997 − 0.0672i)13-s + (0.244 + 0.969i)15-s + (−0.963 + 0.266i)16-s + (0.795 − 0.605i)17-s + (−0.809 + 0.587i)18-s + (−0.156 − 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.482 + 0.875i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.482 + 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.779549258 + 1.050972480i\)
\(L(\frac12)\) \(\approx\) \(1.779549258 + 1.050972480i\)
\(L(1)\) \(\approx\) \(1.181952970 + 0.5014230195i\)
\(L(1)\) \(\approx\) \(1.181952970 + 0.5014230195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.657 + 0.753i)T \)
3 \( 1 + (0.993 + 0.111i)T \)
5 \( 1 + (0.351 + 0.936i)T \)
11 \( 1 + (0.244 - 0.969i)T \)
13 \( 1 + (0.997 - 0.0672i)T \)
17 \( 1 + (0.795 - 0.605i)T \)
19 \( 1 + (-0.156 - 0.987i)T \)
23 \( 1 + (-0.473 + 0.880i)T \)
29 \( 1 + (-0.0224 - 0.999i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (-0.998 - 0.0448i)T \)
47 \( 1 + (0.0672 + 0.997i)T \)
53 \( 1 + (0.795 + 0.605i)T \)
59 \( 1 + (0.0448 - 0.998i)T \)
61 \( 1 + (0.880 - 0.473i)T \)
67 \( 1 + (-0.891 + 0.453i)T \)
71 \( 1 + (0.999 + 0.0224i)T \)
73 \( 1 + (0.974 + 0.222i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.0672 - 0.997i)T \)
97 \( 1 + (-0.891 + 0.453i)T \)
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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

−19.85334715584580871737624902370, −19.29067196053238073873118792632, −18.2531082984816095853366731890, −18.07974402032617690980484436080, −16.770806541480052422359080188506, −16.53076376988522278765919348559, −15.49826816502118329395676852065, −14.58732850073148983274028165688, −13.75852318643131428679296709106, −13.06027889754474431496293427733, −12.41015653399113028269538105962, −11.981341694651782270609690277440, −10.59089744848572730220760254867, −10.013202764008867931078110093716, −9.39733310987129432675833088562, −8.48634364570618023276124953584, −8.28373127049471370345779337379, −7.30912829495490288028902800661, −6.33590785292183935799168726097, −5.07597614956904072558221350161, −3.945046911619566037262944244518, −3.71614104171860431307297241957, −2.31723704994687743515149536669, −1.73752623983225632338766457184, −1.01683493299141622968171758981, 1.003487444089552648582908685447, 1.9618112286145632371997412722, 3.02152891759956194039524787520, 3.657498304177409328557358993631, 4.909828083524111601441891404479, 5.89982342431336612283754863866, 6.58921359545845961075655514735, 7.333963631690256496919802331, 8.117252357864416479115474991241, 8.75044310105813493838681415801, 9.542230521393853793713653584324, 10.145361634395032761520424761517, 10.93691102683544470624386543062, 11.650236615207779953478449482161, 13.25352553171742668470476514577, 13.867700840675146109030072520596, 14.09704136774012824277629141551, 15.10355636115100169545250570376, 15.639060664498689675367046706941, 16.225130991143335461636331402577, 17.252101866824773770340839227503, 17.99356073109381159432674430516, 18.70010395608328959810871130644, 19.11707143526818672839369919296, 19.778543262615532597539821140186

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