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Properties

Label 1-2015-2015.123-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.829 + 0.558i$
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  + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s − i·12-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + 18-s + (−0.866 − 0.5i)19-s − i·21-s + (−0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)6-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s − i·12-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + 18-s + (−0.866 − 0.5i)19-s − i·21-s + (−0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.829 + 0.558i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.829 + 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3718974806 + 1.219089416i\)
\(L(\frac12)\) \(\approx\) \(0.3718974806 + 1.219089416i\)
\(L(1)\) \(\approx\) \(0.6020862973 + 0.5950352822i\)
\(L(1)\) \(\approx\) \(0.6020862973 + 0.5950352822i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−19.40236027889085641254803754666, −18.7353451658700982988319613836, −18.33393778461923779782497183855, −17.29456646935205918784665722555, −16.649060254648090512236233208220, −15.93395641470676932892631720763, −14.93039752502658572908289822854, −13.95944295969245946271722556065, −13.394392273345915191702105635059, −12.76872122423324249015495396244, −12.21620799734318172166992507895, −11.21778620085422016897300447449, −10.72110630257542416013506806269, −10.16949779122879723099709407623, −9.28357099815649575184156459454, −8.08930621440140570110539957106, −7.221792316021855289853963675538, −6.415796812064550236495317255136, −5.52594970230769989938211914220, −5.02800391106978756301952408333, −3.95449412655829307274403831688, −3.193600431877522115891940042172, −2.16397309143213139982133877163, −1.112219530980554205073226613274, −0.44115036701188311578278080409, 0.51169590108086010912214700362, 2.27350362382089245333730823992, 3.241496173123513193564643288005, 4.126149262875413404567391699669, 5.07631436151041529077877223746, 5.442435539473205819393043888519, 6.38229051748681455039531090445, 6.86036687529462472696453299350, 7.99030658576698768022279834976, 8.73039492785491387900053382635, 9.65646398008299291630968451228, 10.26546629298312663859501983477, 11.3801034635087512888572204588, 12.068786639080605722555573647160, 12.880787605846750715221598248231, 13.13636199829485811431323494620, 14.54619803841627474512280016735, 15.32012712112758806254509591044, 15.40070721548561086473802650019, 16.43780800453326559799243814118, 16.95357647311807543483528149976, 17.60233062318377344091560297695, 18.5685349797084434560139697358, 18.84923359645504532000388824252, 20.35009474972214052013895862327

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