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Properties

Label 1-2015-2015.309-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.859 - 0.511i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 12-s + 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + (0.5 + 0.866i)19-s + 21-s + (−0.5 − 0.866i)22-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 12-s + 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + (0.5 + 0.866i)19-s + 21-s + (−0.5 − 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3874321783 - 0.1064792645i\)
\(L(\frac12)\) \(\approx\) \(0.3874321783 - 0.1064792645i\)
\(L(1)\) \(\approx\) \(0.4884284977 + 0.2737039849i\)
\(L(1)\) \(\approx\) \(0.4884284977 + 0.2737039849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−19.67079836163690317819882094535, −19.032095342238528400929857720424, −18.334846836239782485591427041954, −18.0320904955183000038300647919, −17.09213285707749495857636831901, −16.371731860515257077096035447987, −15.744894277341278387811436036119, −14.49856052150626320601034061175, −13.50332666485122923703510581622, −13.028677192541022803297279686134, −12.40126746213153827826642640933, −11.66855153024161477548271596392, −11.03739236732555675285932321397, −10.35284268963359761002288093101, −9.359243272991989776483129892810, −8.453474647266963566184550470927, −8.148291318921721475835817280040, −6.96538069081380673531899938539, −6.24162431539665451460065198326, −5.364200009038172673572907079997, −4.445741593647938159258553267667, −3.08088622867399528108658775058, −2.636390988371524333948512394685, −1.67087801350387524554001382679, −0.66585244032251173586986622276, 0.15132436876953340618828899626, 1.06379577831161452718295479265, 2.50768684906995225872586202055, 3.94514313794125488698347773731, 4.31183304156848840601025296997, 5.42052673278534550485088931220, 5.86640250891472677149372448872, 7.07393059064453038151462237832, 7.350160055536586695886876024139, 8.50072701144885708897381967169, 9.41507969884568756491019413850, 10.00138304929184850651478939271, 10.37954055997056214265823711441, 11.34705543512209593390212210826, 12.240857918134824563232067341572, 13.359654841151766918373407817553, 13.97949584605204713777503583697, 14.80787189859353621656097952595, 15.64096753656341944284197180062, 16.02186307064208826122947547632, 16.67817934575877572526187829525, 17.519741181056173849312296039202, 17.84665699528652500501656636126, 18.776875046294077529235713079851, 19.78642856797107229715879089915

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