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Properties

Label 1-2015-2015.197-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.192 + 0.981i$
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  + (−0.104 − 0.994i)2-s + (0.587 − 0.809i)3-s + (−0.978 + 0.207i)4-s + (−0.866 − 0.5i)6-s + (−0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.309 − 0.951i)9-s + (0.207 + 0.978i)11-s + (−0.406 + 0.913i)12-s + (−0.669 + 0.743i)14-s + (0.913 − 0.406i)16-s + (0.207 − 0.978i)17-s + (−0.913 + 0.406i)18-s + (0.994 − 0.104i)19-s + (−0.994 + 0.104i)21-s + (0.951 − 0.309i)22-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.587 − 0.809i)3-s + (−0.978 + 0.207i)4-s + (−0.866 − 0.5i)6-s + (−0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.309 − 0.951i)9-s + (0.207 + 0.978i)11-s + (−0.406 + 0.913i)12-s + (−0.669 + 0.743i)14-s + (0.913 − 0.406i)16-s + (0.207 − 0.978i)17-s + (−0.913 + 0.406i)18-s + (0.994 − 0.104i)19-s + (−0.994 + 0.104i)21-s + (0.951 − 0.309i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.192 + 0.981i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.192 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7659095777 - 0.9306931799i\)
\(L(\frac12)\) \(\approx\) \(-0.7659095777 - 0.9306931799i\)
\(L(1)\) \(\approx\) \(0.6039381621 - 0.7789351168i\)
\(L(1)\) \(\approx\) \(0.6039381621 - 0.7789351168i\)

Euler product

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

−20.06118811449596288394871616176, −19.54502171525920987905110016252, −18.630521277859405673146783819024, −18.293845457487021889866963438377, −16.91779188513212839494862319443, −16.529490048091735695221381394375, −15.92902728411000734360835571327, −15.22877169240609950425714890883, −14.59362433214463202265877157015, −13.95999409370174704365894321820, −13.174542889958206971902120312416, −12.413375542552468061948113021170, −11.25227474570026925712694692764, −10.28934361717256897113916647044, −9.69380998524027576124932149567, −8.85244700230853425732556140000, −8.48822321993835006903198045715, −7.65285986025433619846659235062, −6.5513290490191290877446325589, −5.81288008182066295140020491127, −5.217479504205246711255436539008, −4.147987574888310091190568421238, −3.44659338005903561029740888286, −2.64157427614226901357162598078, −1.136291007041361949365105532757, 0.23745679849640294612544541814, 1.0197655180412556974551373689, 1.917002602112565535469266867521, 2.80118265788253969277347685254, 3.51575376379778918683318072655, 4.28680707610494424662236541221, 5.356747041439421151097034694364, 6.48207468752474745168530668876, 7.522494552149645098046837876752, 7.701590121065717117532738269073, 9.054578770853084788121382720711, 9.58904359050366450584944893380, 10.06426733525330901242823400064, 11.29378093779134018940790798926, 11.91829151785209018127506009327, 12.56623400932807817709983138732, 13.41862008158683598776629893646, 13.72560879033458429069310602884, 14.513599857154988218622125358914, 15.46285179526855354901671795587, 16.50709656418101110307177772422, 17.458163896243011308912921840896, 17.890841794390714246189986125, 18.6792061696548625449458589613, 19.38553960175304655635098040641

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