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Properties

Label 1-3015-3015.1868-r0-0-0
Degree $1$
Conductor $3015$
Sign $-0.740 + 0.671i$
Analytic cond. $14.0016$
Root an. cond. $14.0016$
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  + (0.690 + 0.723i)2-s + (−0.0475 + 0.998i)4-s + (−0.690 − 0.723i)7-s + (−0.755 + 0.654i)8-s + (0.995 + 0.0950i)11-s + (−0.189 + 0.981i)13-s + (0.0475 − 0.998i)14-s + (−0.995 − 0.0950i)16-s + (0.540 + 0.841i)17-s + (0.959 − 0.281i)19-s + (0.618 + 0.786i)22-s + (−0.371 − 0.928i)23-s + (−0.841 + 0.540i)26-s + (0.755 − 0.654i)28-s + (−0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.690 + 0.723i)2-s + (−0.0475 + 0.998i)4-s + (−0.690 − 0.723i)7-s + (−0.755 + 0.654i)8-s + (0.995 + 0.0950i)11-s + (−0.189 + 0.981i)13-s + (0.0475 − 0.998i)14-s + (−0.995 − 0.0950i)16-s + (0.540 + 0.841i)17-s + (0.959 − 0.281i)19-s + (0.618 + 0.786i)22-s + (−0.371 − 0.928i)23-s + (−0.841 + 0.540i)26-s + (0.755 − 0.654i)28-s + (−0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3015\)    =    \(3^{2} \cdot 5 \cdot 67\)
Sign: $-0.740 + 0.671i$
Analytic conductor: \(14.0016\)
Root analytic conductor: \(14.0016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3015} (1868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3015,\ (0:\ ),\ -0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7342509288 + 1.902305695i\)
\(L(\frac12)\) \(\approx\) \(0.7342509288 + 1.902305695i\)
\(L(1)\) \(\approx\) \(1.175212698 + 0.7614049013i\)
\(L(1)\) \(\approx\) \(1.175212698 + 0.7614049013i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.690 + 0.723i)T \)
7 \( 1 + (-0.690 - 0.723i)T \)
11 \( 1 + (0.995 + 0.0950i)T \)
13 \( 1 + (-0.189 + 0.981i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-0.371 - 0.928i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.327 + 0.945i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.888 + 0.458i)T \)
43 \( 1 + (-0.998 + 0.0475i)T \)
47 \( 1 + (0.618 - 0.786i)T \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (-0.0950 + 0.995i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−18.91302010988843357537881696033, −18.37137724319239197450893915174, −17.52071316990706219758963860794, −16.593771225611692610284817108826, −15.72106284968869182213337224107, −15.277214302369551117782284417132, −14.47044030210064455235774727764, −13.75709084854139678775643654186, −13.1419292911765809592736868165, −12.32130872054500952271008184463, −11.80746526014918128923820587294, −11.29433084751081677090345271031, −10.16204968740318649328038067604, −9.57036884315240219887417670709, −9.20624226548762684696358257254, −7.968645153008945683242054305767, −7.11264967510691082334494719747, −6.04330567997470303606074767623, −5.71033857516178131308123567146, −4.88513149459928616485667450968, −3.79411549699373546016586873932, −3.25229552815870392537316269594, −2.52671240529874469546083730969, −1.52467472558900319271040101466, −0.52322081640422643508779431318, 1.13162920779621334358251841766, 2.285609368913229764510330532811, 3.45036864516258663836115693280, 3.84515947553011940715373111536, 4.6235698760047050167327617534, 5.56359218917770795532546464366, 6.380733155334692985470895084090, 6.97059864515836682111314007765, 7.456238425372403599432467550532, 8.57233030088680561165391795111, 9.16507953729807579126358203923, 9.9800554666898055090488480576, 10.928273613356295379140023911027, 11.81187065187707731230764460942, 12.39664678556273246190313043677, 13.06285758871698464606020069895, 13.92391505413645185399600861731, 14.36001226347521731065434029832, 14.938413420060404722524616952907, 16.01356434789660809009678596356, 16.571301136667242016351031206610, 16.81600148535558478225721988115, 17.78088521882097914500432349410, 18.46142533165197124686983342403, 19.6012788128649152274053857480

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