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Properties

Label 1-2015-2015.362-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.470 + 0.882i$
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  + (0.978 + 0.207i)2-s + (0.951 + 0.309i)3-s + (0.913 + 0.406i)4-s + (0.866 + 0.5i)6-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (0.809 + 0.587i)9-s + (−0.406 + 0.913i)11-s + (0.743 + 0.669i)12-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + (0.406 + 0.913i)17-s + (0.669 + 0.743i)18-s + (−0.207 + 0.978i)19-s + (0.207 − 0.978i)21-s + (−0.587 + 0.809i)22-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.951 + 0.309i)3-s + (0.913 + 0.406i)4-s + (0.866 + 0.5i)6-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (0.809 + 0.587i)9-s + (−0.406 + 0.913i)11-s + (0.743 + 0.669i)12-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + (0.406 + 0.913i)17-s + (0.669 + 0.743i)18-s + (−0.207 + 0.978i)19-s + (0.207 − 0.978i)21-s + (−0.587 + 0.809i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.470 + 0.882i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.470 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.029788915 + 5.052203603i\)
\(L(\frac12)\) \(\approx\) \(3.029788915 + 5.052203603i\)
\(L(1)\) \(\approx\) \(2.380232191 + 1.011514826i\)
\(L(1)\) \(\approx\) \(2.380232191 + 1.011514826i\)

Euler product

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

−19.53671116166665393364468505210, −18.94470763871206943584867472697, −18.47474478084689227158339822666, −17.380657636069378095091185333135, −16.04911240067800631287604130288, −15.74500340706324608573269508323, −15.059923070168554387937836819107, −14.19523993027039173350066097463, −13.71460978553283440624310590486, −12.99268709656497735992923839949, −12.333740507530267067356473323383, −11.56670046781370519029810151220, −10.838758921197969661088653618752, −9.697684164681652725876031605813, −9.105463337949704952092230822318, −8.18050352396795786834587984795, −7.36249312682016408111580051434, −6.599484314316056644392647330240, −5.61451491909791467960177300846, −5.07853609326154189812344504261, −3.82763432940026289732221244282, −3.17867258957544366353705823240, −2.49193171514711723483404834917, −1.762724628905405465311431818087, −0.51806864125985289012005281993, 1.41410140656207946094469940650, 2.15911437327801265018660431358, 3.1278407816218196233903824959, 4.02464659189321480719146145421, 4.30023486477390534797750081955, 5.3611795189182311918007764800, 6.37204441518772883344730318768, 7.2706280273951310195024891065, 7.79507873388128972515230050205, 8.49612285108654571240954101164, 9.816031408225970155811174929034, 10.33233341936144952944109327966, 10.981461774528034754935584448743, 12.3192651308750295446427397982, 12.73796343085006811048175675482, 13.54147896214044696829329571175, 14.144598391301880244251632860, 14.91066622337248371427655314383, 15.21447346483511942699026073409, 16.43478553504066292868945406420, 16.59901178218706073601338783704, 17.68924027650752793246746745876, 18.74980996209016068476321808475, 19.63064049718049930692779771112, 20.15568904847601600914506175569

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