Properties

Label 1-2009-2009.204-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.718 - 0.695i$
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  + (−0.222 + 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (0.222 − 0.974i)11-s + (0.900 − 0.433i)12-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 18-s − 19-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (0.222 − 0.974i)11-s + (0.900 − 0.433i)12-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8627836505 - 0.3493023875i\)
\(L(\frac12)\) \(\approx\) \(0.8627836505 - 0.3493023875i\)
\(L(1)\) \(\approx\) \(0.7659110896 + 0.1746609998i\)
\(L(1)\) \(\approx\) \(0.7659110896 + 0.1746609998i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 - T \)
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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

−19.716785522507912710119314443712, −19.23596593151692987633786914208, −18.62304989565395410116607637333, −17.9619549453346647934957347918, −17.3188566758337289306023089347, −16.94288750772210427680870005718, −15.75419669019654976744715122065, −14.42205942772639116843571158876, −14.09753426974496421757430711175, −13.28653156593671486230356952572, −12.4019028398845736858588871064, −12.01752225237672259988253689116, −11.16015604947138041979639104662, −10.39738593049056136786751045856, −9.953006771447928504513635725402, −8.94819160013201183936941882553, −8.0234331432586542734775148726, −7.159792512783813477807331641884, −6.45071180650253008887897331714, −5.60889878658932975756513910117, −4.611564950947423478602470498442, −3.74481008260461623864629427508, −2.54210439077354211702416037899, −1.93071106802093637307258280965, −1.26743902758132974826673837762, 0.43732370872447354692530694631, 1.20063927843350555140285475585, 2.92272758431990048082274429217, 4.057585422386289079944084411293, 4.69867335607621435253301987507, 5.67799721800066898085326938055, 5.876899446194316879959352270424, 6.72992466291952971122504921784, 8.11678561697339786618421082865, 8.5365600387085255349653076915, 9.32316386957054369729308181224, 10.29548792493234828567448063890, 10.40632032100571122958545315362, 11.83306487537447477885277549352, 12.49939746036222787501088298832, 13.503357276134704476257924965582, 14.03856206608880156642111546867, 14.96305328381919549777331220700, 15.670945096639483642106041990155, 16.40292219813288766606771937548, 16.727276408650354029903238792488, 17.54244636074922064119719002760, 17.98005876269016492982972522755, 18.94614224159159240320085659324, 19.829422254380242674010078312102

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