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Properties

Label 1-2015-2015.1087-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.927 - 0.374i$
Analytic cond. $9.35762$
Root an. cond. $9.35762$
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  + (−0.309 − 0.951i)2-s + (−0.951 − 0.309i)3-s + (−0.809 + 0.587i)4-s + i·6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.809 + 0.587i)9-s + (0.587 + 0.809i)11-s + (0.951 − 0.309i)12-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (0.309 − 0.951i)18-s + (0.951 − 0.309i)19-s + (0.951 − 0.309i)21-s + (0.587 − 0.809i)22-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.951 − 0.309i)3-s + (−0.809 + 0.587i)4-s + i·6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.809 + 0.587i)9-s + (0.587 + 0.809i)11-s + (0.951 − 0.309i)12-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (0.309 − 0.951i)18-s + (0.951 − 0.309i)19-s + (0.951 − 0.309i)21-s + (0.587 − 0.809i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7206283880 - 0.1399233843i\)
\(L(\frac12)\) \(\approx\) \(0.7206283880 - 0.1399233843i\)
\(L(1)\) \(\approx\) \(0.5861763192 - 0.1980270038i\)
\(L(1)\) \(\approx\) \(0.5861763192 - 0.1980270038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.587 - 0.809i)T \)
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

−19.83309244770128693179547498536, −18.99586488970371769960122708536, −18.37068244272539995153811762705, −17.5855689024278613019124480385, −16.95939188546334043885049998271, −16.242237358982597415733005812, −16.02728261133949492142363653191, −15.10704896838846315413260769655, −14.17471054149139212434419196348, −13.45343378109248545047262431960, −12.82749604285374236451963418935, −11.66908312846337577585045235759, −11.05202449857998108414544127320, −10.1398528062300574406851884198, −9.50790121571902626690926834113, −8.93821085925286235971173972882, −7.718445126686747881651240152385, −6.900065467963715842953083226614, −6.50945533677370481253701726828, −5.56063797331745263820680913507, −4.99682361708883492611468840754, −3.94055347265929898265762892603, −3.29554862312320375646061360697, −1.343919196941156457166464247392, −0.54919240193181484453910573488, 0.71405346188925915663337698804, 1.77761730012151916827124027553, 2.53320430463278718987237530911, 3.649657446970076097137281652804, 4.50292393669883423160059238616, 5.275284164210961435757413738959, 6.30638627812729026644379194865, 6.955282412392624193443786704664, 7.942468113195088315026353479606, 8.91556237329862138943902929494, 9.689049537784465590333851635853, 10.2125082326734326549070337589, 11.15568375835075077305681874125, 11.793420546278728816492880561857, 12.357071169664552837970438791257, 13.045488255445055049197020676061, 13.54849092319308986524055402742, 14.85201012429228447154360257677, 15.597549162179673549564891738519, 16.71117733366163500133812252699, 16.965606762206402345875047930573, 18.01334372499932621492643080973, 18.31524320610760954303512212371, 19.20963239266163422098901753205, 19.72388173042013056454589108889

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