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Properties

Label 1-2013-2013.1256-r0-0-0
Degree $1$
Conductor $2013$
Sign $0.707 + 0.706i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.978 + 0.207i)5-s + (0.669 + 0.743i)7-s − 8-s + (0.669 − 0.743i)10-s + (−0.913 + 0.406i)13-s + (0.978 − 0.207i)14-s + (−0.5 + 0.866i)16-s + (−0.669 + 0.743i)17-s + (−0.669 + 0.743i)19-s + (−0.309 − 0.951i)20-s + (0.309 − 0.951i)23-s + (0.913 + 0.406i)25-s + (−0.104 + 0.994i)26-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.978 + 0.207i)5-s + (0.669 + 0.743i)7-s − 8-s + (0.669 − 0.743i)10-s + (−0.913 + 0.406i)13-s + (0.978 − 0.207i)14-s + (−0.5 + 0.866i)16-s + (−0.669 + 0.743i)17-s + (−0.669 + 0.743i)19-s + (−0.309 − 0.951i)20-s + (0.309 − 0.951i)23-s + (0.913 + 0.406i)25-s + (−0.104 + 0.994i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.707 + 0.706i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ 0.707 + 0.706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.502584315 + 0.6221074510i\)
\(L(\frac12)\) \(\approx\) \(1.502584315 + 0.6221074510i\)
\(L(1)\) \(\approx\) \(1.293449284 - 0.2595450075i\)
\(L(1)\) \(\approx\) \(1.293449284 - 0.2595450075i\)

Euler product

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

−20.13799262499014389579493730840, −18.9871300483118058252305631915, −17.905347804570510292758988274616, −17.54291109270552090362280251977, −17.00965614857754603052832959264, −16.3403972634912086131895174124, −15.20568614705668637959564702340, −14.85213669435023568747134745630, −13.84609178401693607013102764012, −13.479575078168937248896861246891, −12.84722548626614949123791305477, −11.84288925837213193170665024308, −11.01405927936474151738717370395, −10.04110407376821380685325206897, −9.22762296854022542824491626786, −8.610385852448893417470699812082, −7.368381217880304939732672385457, −7.23765634857926908689664905534, −6.082521793739690448614005969908, −5.31750662349765016779394460290, −4.735389923124559511441195830232, −3.935383552610259414271710701200, −2.73174671862679948217366595743, −1.91145502505472724905603632843, −0.41500058747587434306406184580, 1.51469953550214603531155353471, 2.00844434249448240834233596479, 2.712709717729243081705801632931, 3.78920585569988275645564041024, 4.866007883153163507951193011503, 5.27546917504221615914939985458, 6.23473204550670357165192900953, 6.88748759751817610171565802742, 8.43267251196225415136066376635, 8.885822913812864642528233741072, 9.81466905712955042141363259758, 10.46758323412672962076960574199, 11.10925040505115470213638350286, 12.03202707125956285513748431473, 12.62527115304046983379656944055, 13.3051093227393787760249257386, 14.21883753056824438271566448301, 14.78528133682500389311230546227, 15.13061789454044021354835291870, 16.57315915625334962906788735960, 17.28082327659186262167507777147, 18.08992973425315091167288415987, 18.60459719499762846188395253354, 19.27320015602063202683930248951, 20.23295525543577270330029274725

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