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Properties

Label 1-2013-2013.1148-r0-0-0
Degree $1$
Conductor $2013$
Sign $-0.445 + 0.895i$
Analytic cond. $9.34833$
Root an. cond. $9.34833$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + i·10-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s + i·23-s + (−0.809 − 0.587i)25-s + (−0.587 − 0.809i)26-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + i·10-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s + i·23-s + (−0.809 − 0.587i)25-s + (−0.587 − 0.809i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.445 + 0.895i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ -0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4093109454 + 0.6609151755i\)
\(L(\frac12)\) \(\approx\) \(0.4093109454 + 0.6609151755i\)
\(L(1)\) \(\approx\) \(0.7063360079 + 0.1682893236i\)
\(L(1)\) \(\approx\) \(0.7063360079 + 0.1682893236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (-0.951 + 0.309i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.951 - 0.309i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 - iT \)
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

−19.735014381636459359429932945519, −18.81026781442601085580086153212, −18.224197789491874093669951856870, −17.54624379346688848060257469268, −17.21277864877120084191711605788, −16.15061997809252274937237083380, −15.35178933038958735869978014364, −14.788782866415235647000429065121, −13.69478870606839031878777209259, −13.20710600146084000697451787685, −12.04071293185778385343906603609, −11.26648501609764957249256241027, −10.641642902186558041961601800707, −10.26754736024454063014547832173, −9.352002291709314894146538073140, −8.37668341514624574167188460381, −7.77488708549385251391702500626, −6.918693969128338392079434490119, −6.43478858962815729379074124657, −5.26722343210257494459513822813, −4.05519345614698913251987719585, −3.21957640826741915809570105561, −2.40544342495984887881954941684, −1.52006352422948380795092522194, −0.35673051265102134214025428774, 1.363868537749085711138225817890, 1.72165363676443366820319198150, 2.793549770850902868237631097721, 4.18498939912435426994425335283, 5.244489064253545278878933676455, 5.64923064608427412727882472517, 6.66777961665741077592551745918, 7.525333452991715348118763208214, 8.37349531200307769844376125902, 9.1378566855168009924364621773, 9.25703221775423332299106819334, 10.41409241766995440318310412900, 11.3499938943562054588511724865, 11.85442225227608646723460755009, 12.63920214925950644152578970935, 13.78801974532073672851618835584, 14.35169808804475812555322424725, 15.3435119425315883956853218798, 16.08975064940204097410821588359, 16.37480023222624332085094164291, 17.55627915031083801258473203642, 17.77530572649171340453001432138, 18.612540255200721843374416083993, 19.34459492567597205668488026082, 20.16147654248276476240277892975

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