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Properties

Label 1-2009-2009.1103-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.611 - 0.790i$
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  + (−0.280 − 0.959i)2-s + (−0.733 + 0.680i)3-s + (−0.842 + 0.538i)4-s + (−0.599 − 0.800i)5-s + (0.858 + 0.512i)6-s + (0.753 + 0.657i)8-s + (0.0747 − 0.997i)9-s + (−0.599 + 0.800i)10-s + (−0.646 − 0.762i)11-s + (0.251 − 0.967i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (0.420 − 0.907i)16-s + (0.887 − 0.460i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + ⋯
L(s)  = 1  + (−0.280 − 0.959i)2-s + (−0.733 + 0.680i)3-s + (−0.842 + 0.538i)4-s + (−0.599 − 0.800i)5-s + (0.858 + 0.512i)6-s + (0.753 + 0.657i)8-s + (0.0747 − 0.997i)9-s + (−0.599 + 0.800i)10-s + (−0.646 − 0.762i)11-s + (0.251 − 0.967i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (0.420 − 0.907i)16-s + (0.887 − 0.460i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.611 - 0.790i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.611 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4387765710 - 0.2152799902i\)
\(L(\frac12)\) \(\approx\) \(0.4387765710 - 0.2152799902i\)
\(L(1)\) \(\approx\) \(0.4864466470 - 0.1824964236i\)
\(L(1)\) \(\approx\) \(0.4864466470 - 0.1824964236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.280 - 0.959i)T \)
3 \( 1 + (-0.733 + 0.680i)T \)
5 \( 1 + (-0.599 - 0.800i)T \)
11 \( 1 + (-0.646 - 0.762i)T \)
13 \( 1 + (-0.691 + 0.722i)T \)
17 \( 1 + (0.887 - 0.460i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.163 + 0.986i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.251 - 0.967i)T \)
43 \( 1 + (0.858 + 0.512i)T \)
47 \( 1 + (-0.280 - 0.959i)T \)
53 \( 1 + (-0.842 + 0.538i)T \)
59 \( 1 + (0.0149 - 0.999i)T \)
61 \( 1 + (-0.772 + 0.635i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (-0.963 - 0.266i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.971 - 0.237i)T \)
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

−19.63022117728480850336061853533, −18.94156579705588739816336903801, −18.601427173284512460157636794154, −17.76400949544142558787808569603, −17.21362193172553110483443930622, −16.54811025172890984307281696133, −15.64381579789603218044363791509, −14.9867914617169520711855289116, −14.48605548640863366187661438306, −13.45846422481503528319623580191, −12.62935499058502361228938903638, −12.18262251112979488038582674203, −10.94976185660403814286913968888, −10.379013126476033059083511869188, −9.80704354613795999227149234034, −8.3305205451998013359349076975, −7.77227187208488132899999660275, −7.35531711719043945586280789016, −6.40472433930432488601013272542, −5.924223776132263845469855270706, −4.889222964476508818178035780611, −4.2541720094563201511838075337, −2.87808122656022599847557772192, −1.84405943832959566286343334703, −0.46199233808993050266433187010, 0.48560361038004450245192213472, 1.51281715632482608530266823531, 2.8095956789538301424538733341, 3.707800248504761146732132033274, 4.38129336786481715366711777953, 5.11928994154530369826797349547, 5.766776351231523913931426450871, 7.20228015879849534740657563398, 8.016689230240337024203925863389, 8.9044754326376905258118724538, 9.478621453002685797418094512, 10.22324634950600884858116746276, 11.14211341393880206769008147474, 11.52361713286626531749776477804, 12.330077332046943632644927231506, 12.83514604011823053805112134215, 13.817934807952910473132967513014, 14.72567619275582739755271200401, 15.7457534124910259278679656825, 16.38337298266898362177011666839, 16.94555946645609383208116913440, 17.55362239622514707147461124641, 18.54290805603107057452607002840, 19.1898160584534709242058584834, 19.82590334549143682246137332622

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