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Properties

Label 1-2013-2013.1214-r0-0-0
Degree $1$
Conductor $2013$
Sign $-0.997 - 0.0698i$
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  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.669 − 0.743i)5-s + (−0.743 − 0.669i)7-s + (0.951 − 0.309i)8-s + (0.406 + 0.913i)10-s + (0.669 − 0.743i)13-s + (0.913 − 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.669 − 0.743i)19-s − 20-s + (−0.587 − 0.809i)23-s + (−0.104 − 0.994i)25-s + (0.406 + 0.913i)26-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.669 − 0.743i)5-s + (−0.743 − 0.669i)7-s + (0.951 − 0.309i)8-s + (0.406 + 0.913i)10-s + (0.669 − 0.743i)13-s + (0.913 − 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.669 − 0.743i)19-s − 20-s + (−0.587 − 0.809i)23-s + (−0.104 − 0.994i)25-s + (0.406 + 0.913i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.997 - 0.0698i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ -0.997 - 0.0698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002815713118 - 0.08056151902i\)
\(L(\frac12)\) \(\approx\) \(0.002815713118 - 0.08056151902i\)
\(L(1)\) \(\approx\) \(0.6733045512 + 0.05768709618i\)
\(L(1)\) \(\approx\) \(0.6733045512 + 0.05768709618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
5 \( 1 + (0.669 - 0.743i)T \)
7 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (-0.207 + 0.978i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (-0.994 + 0.104i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.994 - 0.104i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.994 + 0.104i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.994 + 0.104i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.207 + 0.978i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.32581137891531367004640989089, −19.33720272621193498510102056648, −18.87038851271226128238658534814, −18.245970477851073703271057012854, −17.76101794422220117117166969032, −16.67505147670169455196631144403, −16.19109723348400417331098948296, −15.11363503016532729072610481254, −14.24222917058187574443369807826, −13.482230266626259761287888372214, −13.02246212372749744594153197620, −11.91268567781336616509741887093, −11.52544940782846999144683014158, −10.509355806477786774737516326333, −9.99132535965337589436159396955, −9.15065540392275184950324851284, −8.75953833453960728205925161790, −7.51470424218495396947770786865, −6.74238568944100506222259385331, −5.87483931826813313229906237341, −5.01481473006411622607530755618, −3.60474425202028153736875527283, −3.357146439607763166886991885513, −2.04209657733558509909733851117, −1.826277351776022965573026850977, 0.032559159755440357636124517729, 1.11621523612392388949135264213, 2.06884819369985233982396571471, 3.57079998605158346687010699779, 4.34403297190417863353247097795, 5.27206550484578653757664957219, 6.06632047149275757319213134614, 6.57247589883093028395286770005, 7.509121789654808001366111665476, 8.558508265447675514969602110911, 8.78852325529759783243865711225, 9.99656648145648256081037482289, 10.22612113620823093171747348065, 11.1884658632776550217510948282, 12.69887495290558630959269699593, 13.117421729460438822565417555940, 13.59477191596155938590322360762, 14.61171271908794794975424176069, 15.30037309386527960926870163695, 16.145640551976381584006454802701, 16.714784138972311773350647703036, 17.23015262526249781723899476060, 17.94879139772444565525172342362, 18.67441397843119560124139301734, 19.620052467062379083221938317312

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