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Properties

Label 1-2009-2009.143-r1-0-0
Degree $1$
Conductor $2009$
Sign $0.903 - 0.428i$
Analytic cond. $215.897$
Root an. cond. $215.897$
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  + (−0.251 + 0.967i)2-s + (−0.997 + 0.0747i)3-s + (−0.873 − 0.486i)4-s + (0.791 + 0.611i)5-s + (0.178 − 0.983i)6-s + (0.691 − 0.722i)8-s + (0.988 − 0.149i)9-s + (−0.791 + 0.611i)10-s + (0.894 + 0.447i)11-s + (0.907 + 0.420i)12-s + (−0.266 − 0.963i)13-s + (−0.834 − 0.550i)15-s + (0.525 + 0.850i)16-s + (−0.999 − 0.0149i)17-s + (−0.104 + 0.994i)18-s + (−0.994 + 0.104i)19-s + ⋯
L(s)  = 1  + (−0.251 + 0.967i)2-s + (−0.997 + 0.0747i)3-s + (−0.873 − 0.486i)4-s + (0.791 + 0.611i)5-s + (0.178 − 0.983i)6-s + (0.691 − 0.722i)8-s + (0.988 − 0.149i)9-s + (−0.791 + 0.611i)10-s + (0.894 + 0.447i)11-s + (0.907 + 0.420i)12-s + (−0.266 − 0.963i)13-s + (−0.834 − 0.550i)15-s + (0.525 + 0.850i)16-s + (−0.999 − 0.0149i)17-s + (−0.104 + 0.994i)18-s + (−0.994 + 0.104i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.903 - 0.428i$
Analytic conductor: \(215.897\)
Root analytic conductor: \(215.897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (1:\ ),\ 0.903 - 0.428i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6230321588 - 0.1403862301i\)
\(L(\frac12)\) \(\approx\) \(0.6230321588 - 0.1403862301i\)
\(L(1)\) \(\approx\) \(0.6004812520 + 0.3179662547i\)
\(L(1)\) \(\approx\) \(0.6004812520 + 0.3179662547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.251 + 0.967i)T \)
3 \( 1 + (-0.997 + 0.0747i)T \)
5 \( 1 + (0.791 + 0.611i)T \)
11 \( 1 + (0.894 + 0.447i)T \)
13 \( 1 + (-0.266 - 0.963i)T \)
17 \( 1 + (-0.999 - 0.0149i)T \)
19 \( 1 + (-0.994 + 0.104i)T \)
23 \( 1 + (-0.599 - 0.800i)T \)
29 \( 1 + (0.0896 + 0.995i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.420 - 0.907i)T \)
43 \( 1 + (-0.983 - 0.178i)T \)
47 \( 1 + (-0.967 - 0.251i)T \)
53 \( 1 + (-0.486 + 0.873i)T \)
59 \( 1 + (0.337 + 0.941i)T \)
61 \( 1 + (0.992 + 0.119i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (-0.0896 + 0.995i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.701 + 0.712i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−19.61262641762984156961943232552, −19.228142982956615990740508945867, −18.28336884490768070718743580821, −17.559022762572361394120331145288, −17.09766740497084020377550335299, −16.62384605656795238761322613001, −15.67340683052854267168246349799, −14.40472096762432221740745017407, −13.57570688884544416680388601461, −13.12101040285796252545684497053, −12.26497577978009743051494960548, −11.5492981965651834775464570930, −11.19396241499135065110030823324, −9.99823692502538173539818788939, −9.690746021858405285577611978598, −8.79858176209527654899450136602, −8.056875780504738697637578548812, −6.59116280913392169773265574355, −6.27535135472271937747048972225, −5.01172325683159498050431296317, −4.526694708240758773075844282369, −3.720406722860710678398916216226, −2.2128777348864812288654932145, −1.69081930102554170788769847466, −0.76675325724238926863023130499, 0.189246363813191283385262173500, 1.285147730307547024928060038, 2.31126765391021598807382423167, 3.82936965062199565703773891679, 4.635982157983069752758054291164, 5.393122778813997543086185363616, 6.318261567469788931647456938545, 6.558611289834325619354449066033, 7.34571287707216024516572002613, 8.43732478994883962911504353484, 9.29887615427674595478673150392, 10.15107946793095979227071365117, 10.48743353012135030852972647467, 11.423400458564225654044609964500, 12.582943807963927109557719636106, 13.06009439561576492021081123563, 14.02518087255007626891715987921, 14.84205624482618393328149010523, 15.25732233776334765686228115168, 16.24774485706738891654682487421, 16.90292436502758979862249951116, 17.67361398812689541513179731260, 17.79281292299594496418900550090, 18.63081836733609473194288157745, 19.46525945151280842767286136324

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