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Properties

Label 1-2009-2009.4-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.637 - 0.770i$
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  + (0.998 − 0.0598i)2-s + (−0.0747 − 0.997i)3-s + (0.992 − 0.119i)4-s + (−0.999 − 0.0299i)5-s + (−0.134 − 0.990i)6-s + (0.983 − 0.178i)8-s + (−0.988 + 0.149i)9-s + (−0.999 + 0.0299i)10-s + (−0.887 + 0.460i)11-s + (−0.193 − 0.981i)12-s + (0.550 + 0.834i)13-s + (0.0448 + 0.998i)15-s + (0.971 − 0.237i)16-s + (0.599 − 0.800i)17-s + (−0.978 + 0.207i)18-s + (0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.998 − 0.0598i)2-s + (−0.0747 − 0.997i)3-s + (0.992 − 0.119i)4-s + (−0.999 − 0.0299i)5-s + (−0.134 − 0.990i)6-s + (0.983 − 0.178i)8-s + (−0.988 + 0.149i)9-s + (−0.999 + 0.0299i)10-s + (−0.887 + 0.460i)11-s + (−0.193 − 0.981i)12-s + (0.550 + 0.834i)13-s + (0.0448 + 0.998i)15-s + (0.971 − 0.237i)16-s + (0.599 − 0.800i)17-s + (−0.978 + 0.207i)18-s + (0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.368527042 - 1.113523604i\)
\(L(\frac12)\) \(\approx\) \(2.368527042 - 1.113523604i\)
\(L(1)\) \(\approx\) \(1.603621026 - 0.5109812985i\)
\(L(1)\) \(\approx\) \(1.603621026 - 0.5109812985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.998 - 0.0598i)T \)
3 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (-0.999 - 0.0299i)T \)
11 \( 1 + (-0.887 + 0.460i)T \)
13 \( 1 + (0.550 + 0.834i)T \)
17 \( 1 + (0.599 - 0.800i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.575 - 0.817i)T \)
29 \( 1 + (-0.753 + 0.657i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.193 + 0.981i)T \)
43 \( 1 + (0.134 + 0.990i)T \)
47 \( 1 + (-0.998 + 0.0598i)T \)
53 \( 1 + (-0.992 + 0.119i)T \)
59 \( 1 + (0.791 - 0.611i)T \)
61 \( 1 + (0.420 - 0.907i)T \)
67 \( 1 + (0.104 - 0.994i)T \)
71 \( 1 + (-0.753 - 0.657i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.447 + 0.894i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−20.3165849363155080909287941871, −19.54802452960597410763976870255, −18.86262237657258580984381730208, −17.64400102670604911494862173061, −16.81214287757211704751331796845, −15.95136083270960956803184863808, −15.70612916794575319351883161440, −15.058270137289699857160245959669, −14.38704407106155744889981733846, −13.39094424848207283358634208341, −12.8092303766327049072482912601, −11.71695463526679847586244832409, −11.333517815158281895179118978730, −10.586951360244392280337489179106, −9.93557435834538617177975942861, −8.626409517981673647506731680373, −7.93586572890858947620382038353, −7.29989364009869199591807119629, −5.9397208394502593734465642287, −5.49810959615357576464799978317, −4.70833345425234551455348933624, −3.71256808288699102243186878842, −3.393672047891666701481912219957, −2.550153742325532602869498575835, −0.876785954794566270499788390154, 0.881111692990525781850433716218, 1.831896521862978005048745895815, 2.952108877840269375082356158708, 3.383291636559801838533175614950, 4.74975683096035722988685286157, 5.09176501314608439340502962211, 6.32935579057659118788664758668, 6.8840478474602592032577965426, 7.70668626762681381889478917882, 8.104162799369002960919370482835, 9.3636656406661655974652753261, 10.579580419924055245572120765315, 11.35772365898852265646646748051, 11.82624927622499294965893579682, 12.52556333818120853166948941467, 13.12617897324875160059393961766, 13.93177306871597558961462033120, 14.545513129298318270651579520173, 15.34333932084214609041305942496, 16.292312742818446896541673932775, 16.496985539533570780451943059578, 17.82259297107110057500472053750, 18.6997081083370938704864691332, 19.00779175417844743862598873351, 19.99576336995072046755840306798

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