Properties

Label 1-2009-2009.155-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: $\chi_{2009} (155, \cdot )$
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

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

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