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Properties

Label 1-2013-2013.1205-r0-0-0
Degree $1$
Conductor $2013$
Sign $-0.945 + 0.324i$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (−0.5 − 0.866i)7-s + (0.809 + 0.587i)8-s + (−0.104 − 0.994i)10-s + (−0.913 − 0.406i)13-s + (0.978 + 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.978 + 0.207i)17-s + (0.5 − 0.866i)19-s + (0.809 + 0.587i)20-s + (−0.809 − 0.587i)23-s + (−0.104 − 0.994i)25-s + (0.913 − 0.406i)26-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (−0.5 − 0.866i)7-s + (0.809 + 0.587i)8-s + (−0.104 − 0.994i)10-s + (−0.913 − 0.406i)13-s + (0.978 + 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.978 + 0.207i)17-s + (0.5 − 0.866i)19-s + (0.809 + 0.587i)20-s + (−0.809 − 0.587i)23-s + (−0.104 − 0.994i)25-s + (0.913 − 0.406i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.945 + 0.324i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ -0.945 + 0.324i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03463469129 + 0.2075662697i\)
\(L(\frac12)\) \(\approx\) \(0.03463469129 + 0.2075662697i\)
\(L(1)\) \(\approx\) \(0.5105727844 + 0.1368070550i\)
\(L(1)\) \(\approx\) \(0.5105727844 + 0.1368070550i\)

Euler product

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

−19.53140758438181267928912842165, −18.88790079395282123461560295419, −18.51472710457640963842586180803, −17.42694733968999889090604304240, −16.70204096711926155528328732837, −16.18965688240503147576503581896, −15.502960236064288180622793608130, −14.47780247206777262628122084967, −13.530194665082155955147074279232, −12.48911654508824655863874816446, −12.19415222829952960002334553309, −11.74473715718866487973471438920, −10.66737627557557289345337999629, −9.736577387721016528858502452800, −9.26250640134951487363495106476, −8.55356311502234505864430020585, −7.65156460540148446934454468565, −7.19311377038310177187958801800, −5.72697901246583515315953967583, −5.05941265147479088700548673934, −3.85251278248377317295319580059, −3.38591309078303413212624072420, −2.23555900067873894446322941170, −1.44951825600582761560917686733, −0.11896434759483398503623019641, 0.86284177994966787960376256051, 2.252686184961354593768156417505, 3.29178872902606639730290124612, 4.16973607932313168963071698006, 5.13977298929887485269402835026, 6.0800302723977399540123193536, 6.9343621747551998994989033671, 7.465575763940129556123886395623, 7.95745623713709120251329193148, 9.04999885646168637054050949050, 9.99821765281067360555449041611, 10.33184989758355568815610667134, 11.167856806105508931638418814845, 12.047577625580470190641269701557, 13.03298402260469173666621398045, 14.051040695203896386060994517147, 14.46793193508971976692240255734, 15.3028956608079075219102493342, 15.908625743500354838595240388040, 16.672272290678921193073624962489, 17.25894249726372369159257422971, 18.09470626821221839433853948678, 18.74275178533920810600875487400, 19.56508146426355874116056247515, 19.834263338802475367411691662066

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