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Properties

Label 1-2015-2015.219-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.568 + 0.822i$
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  + (−0.207 + 0.978i)2-s + (−0.669 − 0.743i)3-s + (−0.913 − 0.406i)4-s + (0.866 − 0.5i)6-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (−0.104 + 0.994i)9-s + (−0.994 + 0.104i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)14-s + (0.669 + 0.743i)16-s + (−0.104 + 0.994i)17-s + (−0.951 − 0.309i)18-s + (0.743 + 0.669i)19-s + (0.951 − 0.309i)21-s + (0.104 − 0.994i)22-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.669 − 0.743i)3-s + (−0.913 − 0.406i)4-s + (0.866 − 0.5i)6-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (−0.104 + 0.994i)9-s + (−0.994 + 0.104i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)14-s + (0.669 + 0.743i)16-s + (−0.104 + 0.994i)17-s + (−0.951 − 0.309i)18-s + (0.743 + 0.669i)19-s + (0.951 − 0.309i)21-s + (0.104 − 0.994i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.568 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7028266444 + 0.3687055034i\)
\(L(\frac12)\) \(\approx\) \(0.7028266444 + 0.3687055034i\)
\(L(1)\) \(\approx\) \(0.5712705373 + 0.2184181427i\)
\(L(1)\) \(\approx\) \(0.5712705373 + 0.2184181427i\)

Euler product

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

−19.848497051782619387433702500140, −18.910623773167752366425585934540, −18.15707205679896516018452916098, −17.558477832097831170467235937882, −16.81037745892275756076129222806, −16.17804417956071074942846352252, −15.44534835059741383521039952738, −14.38499209950816907525998312274, −13.47566684114070896109322291959, −13.03689347933640592152442329268, −12.07197737615657099669713034937, −11.18175454243132964092479666198, −10.926310516820597928242153367273, −9.97334321048586995077531051548, −9.56926490178892670862973577732, −8.74597052707691607104966890227, −7.564410651058328777213545037081, −6.92915738836761595447117325212, −5.53761821839545898661710099038, −4.98760225889190391166801621933, −4.19684847110398599055407005581, −3.29081097778117369107550553230, −2.72436376963128234417760158404, −1.17890953659265984598765060432, −0.43757507556251337243324888781, 0.4017757564878329949027701607, 1.57894591165467074669544793341, 2.56086453760601120365258509990, 3.84006522707689396677872912283, 5.032477123010144022412508008186, 5.66383192953434592436299222304, 6.04757280122224251555197292428, 7.132806936785914999203476917917, 7.59720959412018237926546197869, 8.51979292146680408644295683772, 9.14056003544203374661764394966, 10.28312653819876071355806997570, 10.77452440262200038362291186934, 12.01588693581161393991866081618, 12.65175473109711477088699533508, 13.1866882080626627256006295794, 14.00216741838833194737691277396, 14.947758986161570507065952700357, 15.62637396589928785437082016682, 16.23042323214111394867250218301, 17.02362700941864103954707040192, 17.63071062252977281992817545291, 18.42102838457997974540498623379, 18.85603508331226021598492557967, 19.34124195809244323955194303157

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