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Properties

Label 1-2009-2009.1649-r0-0-0
Degree $1$
Conductor $2009$
Sign $-0.102 + 0.994i$
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  + (0.988 − 0.149i)2-s + (−0.563 + 0.826i)3-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (−0.433 + 0.900i)6-s + (0.900 − 0.433i)8-s + (−0.365 − 0.930i)9-s + (0.0747 + 0.997i)10-s + (0.930 + 0.365i)11-s + (−0.294 + 0.955i)12-s + (−0.781 + 0.623i)13-s + (−0.781 − 0.623i)15-s + (0.826 − 0.563i)16-s + (0.680 + 0.733i)17-s + (−0.5 − 0.866i)18-s + (0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (−0.563 + 0.826i)3-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (−0.433 + 0.900i)6-s + (0.900 − 0.433i)8-s + (−0.365 − 0.930i)9-s + (0.0747 + 0.997i)10-s + (0.930 + 0.365i)11-s + (−0.294 + 0.955i)12-s + (−0.781 + 0.623i)13-s + (−0.781 − 0.623i)15-s + (0.826 − 0.563i)16-s + (0.680 + 0.733i)17-s + (−0.5 − 0.866i)18-s + (0.866 + 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $-0.102 + 0.994i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ -0.102 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.837451297 + 2.035707812i\)
\(L(\frac12)\) \(\approx\) \(1.837451297 + 2.035707812i\)
\(L(1)\) \(\approx\) \(1.590173309 + 0.6894796693i\)
\(L(1)\) \(\approx\) \(1.590173309 + 0.6894796693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.988 - 0.149i)T \)
3 \( 1 + (-0.563 + 0.826i)T \)
5 \( 1 + (-0.0747 + 0.997i)T \)
11 \( 1 + (0.930 + 0.365i)T \)
13 \( 1 + (-0.781 + 0.623i)T \)
17 \( 1 + (0.680 + 0.733i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (0.974 - 0.222i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.149 + 0.988i)T \)
53 \( 1 + (0.294 + 0.955i)T \)
59 \( 1 + (0.0747 + 0.997i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.974 - 0.222i)T \)
73 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.930 + 0.365i)T \)
97 \( 1 - iT \)
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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

−19.78859092955649820222025569042, −19.40448940366511083423223361619, −18.05352160530607407159615551963, −17.47500725658219992411211868233, −16.64158396277767868218356148008, −16.26007091914351522011832566083, −15.44456119242931067158461014235, −14.253170517028004265253564832493, −13.8827338532843936978921074122, −13.06082007571321039051483313540, −12.325342689265701048840169363385, −11.92312607199768737053308774974, −11.353136610199693736639457793, −10.24602073739888747305697547790, −9.233391587602810172333987303196, −8.15948845901894005837186079710, −7.51217708685121510341681580030, −6.83570257556935699819367732595, −5.821302669557884264739696654679, −5.31506292608496757277180040528, −4.67081395577745088284844660382, −3.58175601218204676789177660193, −2.63109561975604571636808127704, −1.55435904575558733925742827196, −0.76634342407141543706250394852, 1.29367200205750686300346762314, 2.47781294433226281051986340661, 3.272198490543352830967626123288, 4.191535944708643165041604257465, 4.49382774194753848079885857321, 5.920686180899562574524532730085, 6.07137231161404389656669709129, 7.07090345736507457812601783569, 7.80702641986481045060129207509, 9.338138238123355121559923158933, 9.99955451916142155353317578459, 10.55040869449354356403860199182, 11.44127278919284877731689811043, 11.9638146912125239380441567345, 12.47266294024405328146877606431, 13.84689569008331500316361928802, 14.51071647286615453796320539493, 14.76196931415351127216370087975, 15.60209590863712841984329192099, 16.43084577109432202806917613605, 16.95779818044960627010984561952, 17.87295436659275863010991142837, 18.81650024876044878021669675946, 19.61340184887754362023482067706, 20.249052504637155577074756759722

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