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Properties

Label 1-2015-2015.213-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.205 + 0.978i$
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  + (0.809 − 0.587i)2-s + (−0.587 + 0.809i)3-s + (0.309 − 0.951i)4-s + i·6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.587 + 0.809i)12-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + (−0.809 − 0.587i)18-s + (−0.587 − 0.809i)19-s + (0.587 + 0.809i)21-s + (−0.951 + 0.309i)22-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.587 + 0.809i)3-s + (0.309 − 0.951i)4-s + i·6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.587 + 0.809i)12-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + (−0.809 − 0.587i)18-s + (−0.587 − 0.809i)19-s + (0.587 + 0.809i)21-s + (−0.951 + 0.309i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.205 + 0.978i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.205 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07866706393 + 0.06386113904i\)
\(L(\frac12)\) \(\approx\) \(0.07866706393 + 0.06386113904i\)
\(L(1)\) \(\approx\) \(0.9900442648 - 0.4409363446i\)
\(L(1)\) \(\approx\) \(0.9900442648 - 0.4409363446i\)

Euler product

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

−19.45438208138956080236057844402, −18.61191086194441456780655933922, −18.13525640728111987064979544264, −17.35878057966934513029430102708, −16.63014379525618993558410801090, −15.94895912694338535472236483550, −15.168320648423666763821182603149, −14.44608800665013435882418492874, −13.73213424933088209719038967894, −12.77994925153932817037811595856, −12.431591896817077284758346425358, −11.81790863576375903377485610587, −10.97338585085161823981977559828, −10.082623131797111057829543433860, −8.74058621484412468799244269573, −7.88947987956504387810466812108, −7.67006848892853914910651835290, −6.45372585459456498705555456728, −5.856213628350742709415061380681, −5.33120028580222672161223914325, −4.511115177218514903054562816367, −3.339762481063551212953459939837, −2.30616584940785979661012069000, −1.75152541433701634201170179219, −0.01669335553451777939102806485, 0.774604921285257164653561417825, 1.93238585346622409908261751013, 3.17080437333443729277718852872, 3.704992144388393674153198721, 4.64577338467777734298130847736, 5.176791020710703422557532434079, 5.93017254901910066076762077860, 6.85821136169914378429373804022, 7.75554884728623205884527007988, 8.96142105091168014198715050433, 9.922420497336551139427006842224, 10.443421406809932620670323198034, 10.97541360767645739838022331954, 11.70752045123990470927042999419, 12.453774931441817914953013176980, 13.38465187295155061496509772386, 13.93275686822239209897315063217, 14.84411906136577922467027358685, 15.37401670940854878996129956007, 16.31768495028035585196377909440, 16.70512215772956579449852146524, 17.84639941878714045765961257849, 18.38546600553093722205468675607, 19.49284338471736317732120070369, 20.13633350343738711383124652559

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