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Properties

Label 1-2015-2015.74-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.576 - 0.816i$
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  + (−0.669 + 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 − 0.994i)4-s + (0.5 + 0.866i)6-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.104 + 0.994i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (0.978 − 0.207i)18-s + (0.669 − 0.743i)19-s + (−0.669 + 0.743i)21-s + (−0.809 − 0.587i)22-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 − 0.994i)4-s + (0.5 + 0.866i)6-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.104 + 0.994i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (0.978 − 0.207i)18-s + (0.669 − 0.743i)19-s + (−0.669 + 0.743i)21-s + (−0.809 − 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.576 - 0.816i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.576 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1930456163 - 0.3726907477i\)
\(L(\frac12)\) \(\approx\) \(0.1930456163 - 0.3726907477i\)
\(L(1)\) \(\approx\) \(0.6671198331 + 0.002261990557i\)
\(L(1)\) \(\approx\) \(0.6671198331 + 0.002261990557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.669 + 0.743i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.669 - 0.743i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−20.05159557209440040261302882259, −19.31663312901662231890958099046, −18.69681418191246546455888892595, −18.07126409707014062241670354075, −16.842445008027899337297417884674, −16.40294975816779818812800960875, −15.99284152710495202205395644106, −15.02066932276363440589500654124, −14.00515283141134937359239423950, −13.40494554600223699002779297121, −12.5132155578642234817498574706, −11.54996671151860772883317935396, −11.18334137777412122020170224053, −9.93066775290303270616948362746, −9.885328376930912465600852017296, −8.899398476030624937713385355449, −8.40524339889969181151450576714, −7.47731407193336174626480810141, −6.3410107032417980027813611784, −5.45419469106165276586571863525, −4.38318996719580123126698723263, −3.52846765327089335195745091438, −2.97105049417046822737349909822, −2.23080803347947259535610240684, −0.74545050782548929108693878798, 0.12272200105869095419996154891, 1.194612317227184939370670957605, 1.9294012201454683374266183855, 3.048623608225951699723019756185, 4.10756068811144881202637537497, 5.2943843185350373940985968130, 6.1460722803031390515885873383, 6.80924481773387307104410170069, 7.41709463262750570243464300251, 7.97637041616089048379312203951, 9.16274979327528156069631107323, 9.41456538804543140413823627658, 10.41353339655702188008377903460, 11.263952916573076952678021459416, 12.31124737967716444651965806394, 13.031587812452224700727142211, 13.662760792608809290714648589525, 14.42676895718460960395756759589, 15.24316433090840062847124589804, 15.778909657808524884821756955626, 16.89614516206080719571058750328, 17.29517137821141609225730809417, 18.0202165650425805966556360549, 18.72157395157179265311734519825, 19.4136710882367464390666881161

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