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Properties

Label 1-2009-2009.337-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.894 - 0.447i$
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 + (−0.781 + 0.623i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.781 − 0.623i)6-s + (−0.623 − 0.781i)8-s + (0.222 − 0.974i)9-s + (0.623 − 0.781i)10-s + (0.974 − 0.222i)11-s + (0.433 − 0.900i)12-s + (0.974 − 0.222i)13-s + (0.974 + 0.222i)15-s + (0.623 − 0.781i)16-s + (−0.433 + 0.900i)17-s + 18-s − i·19-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.781 + 0.623i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.781 − 0.623i)6-s + (−0.623 − 0.781i)8-s + (0.222 − 0.974i)9-s + (0.623 − 0.781i)10-s + (0.974 − 0.222i)11-s + (0.433 − 0.900i)12-s + (0.974 − 0.222i)13-s + (0.974 + 0.222i)15-s + (0.623 − 0.781i)16-s + (−0.433 + 0.900i)17-s + 18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6741034735 - 0.1592796509i\)
\(L(\frac12)\) \(\approx\) \(0.6741034735 - 0.1592796509i\)
\(L(1)\) \(\approx\) \(0.6762533808 + 0.2884157570i\)
\(L(1)\) \(\approx\) \(0.6762533808 + 0.2884157570i\)

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 \( 1 + (-0.781 + 0.623i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.974 - 0.222i)T \)
13 \( 1 + (0.974 - 0.222i)T \)
17 \( 1 + (-0.433 + 0.900i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.433 - 0.900i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (-0.433 - 0.900i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.433 - 0.900i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.974 - 0.222i)T \)
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

−19.99643664312195463684963944705, −19.05439620559713508238480448852, −18.73005164904158845209364644017, −18.024506351418988680621335116660, −17.448743093355065139535779084929, −16.36592069353124553333632098961, −15.69510621594607529447690882645, −14.51879175033378523142511122233, −14.03465110947287174027702710139, −13.31517573440554693279614919529, −12.27397664051387786721749744551, −11.82071734686923659581947945936, −11.38098478015931518924676465543, −10.49305658169323458082461795233, −9.99985811099593789663176183176, −8.73410981680464219858308607565, −8.089277107189004247479062173667, −6.88983072593378333914672457537, −6.42981148850602332691261255230, −5.49091733756867092334758978916, −4.441722563621909221481964858443, −3.81851764850350106304026102874, −2.8534224752361383608210108015, −1.837488617629848401657092221683, −1.043018761176669430924381697892, 0.312287725280057328917637262092, 1.352416633702246760839659453879, 3.40516957472896010440572291695, 3.98503398008190168469496476755, 4.59961238546301177908112452626, 5.39708790094941097052196778723, 6.29432803097080008790327738185, 6.67528898557336289032444236552, 7.99548758006022043034877540145, 8.55348466427898610887431964921, 9.27079291395806944686393740586, 10.048847663837465522565884572758, 11.24417666188207293965240096189, 11.73880624720890206181572965577, 12.58881995730746799229664322607, 13.28585923534213131971183618548, 14.12899550547969309144176310648, 15.209201633307809842249416482326, 15.56049991517875274149026071629, 16.20032668355349500128546079492, 16.83125761401888760489188509283, 17.55545212934844767004600674996, 17.92855027102541581518350061979, 19.22435727945036063477455691044, 19.77193614154870325008738060617

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