Properties

Label 2-6015-1.1-c1-0-201
Degree $2$
Conductor $6015$
Sign $-1$
Analytic cond. $48.0300$
Root an. cond. $6.93036$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.403·2-s + 3-s − 1.83·4-s − 5-s + 0.403·6-s + 0.330·7-s − 1.54·8-s + 9-s − 0.403·10-s + 0.933·11-s − 1.83·12-s − 4.15·13-s + 0.133·14-s − 15-s + 3.04·16-s − 0.316·17-s + 0.403·18-s + 2.03·19-s + 1.83·20-s + 0.330·21-s + 0.377·22-s + 1.34·23-s − 1.54·24-s + 25-s − 1.67·26-s + 27-s − 0.607·28-s + ⋯
L(s)  = 1  + 0.285·2-s + 0.577·3-s − 0.918·4-s − 0.447·5-s + 0.164·6-s + 0.124·7-s − 0.547·8-s + 0.333·9-s − 0.127·10-s + 0.281·11-s − 0.530·12-s − 1.15·13-s + 0.0356·14-s − 0.258·15-s + 0.761·16-s − 0.0768·17-s + 0.0952·18-s + 0.466·19-s + 0.410·20-s + 0.0721·21-s + 0.0803·22-s + 0.281·23-s − 0.316·24-s + 0.200·25-s − 0.329·26-s + 0.192·27-s − 0.114·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6015\)    =    \(3 \cdot 5 \cdot 401\)
Sign: $-1$
Analytic conductor: \(48.0300\)
Root analytic conductor: \(6.93036\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6015,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
401 \( 1 - T \)
good2 \( 1 - 0.403T + 2T^{2} \)
7 \( 1 - 0.330T + 7T^{2} \)
11 \( 1 - 0.933T + 11T^{2} \)
13 \( 1 + 4.15T + 13T^{2} \)
17 \( 1 + 0.316T + 17T^{2} \)
19 \( 1 - 2.03T + 19T^{2} \)
23 \( 1 - 1.34T + 23T^{2} \)
29 \( 1 - 0.499T + 29T^{2} \)
31 \( 1 - 2.54T + 31T^{2} \)
37 \( 1 + 6.91T + 37T^{2} \)
41 \( 1 - 6.76T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 - 0.790T + 53T^{2} \)
59 \( 1 + 8.46T + 59T^{2} \)
61 \( 1 - 3.59T + 61T^{2} \)
67 \( 1 + 5.69T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 8.44T + 73T^{2} \)
79 \( 1 - 7.10T + 79T^{2} \)
83 \( 1 + 8.85T + 83T^{2} \)
89 \( 1 - 2.47T + 89T^{2} \)
97 \( 1 + 8.94T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.82168365693021670414109571838, −7.17776444276278544734586409081, −6.31274492943501720660573990341, −5.31260740484131249293977572008, −4.73836461054798905439521732166, −4.09820171153821248929258697875, −3.30002488714173353034651158970, −2.57700579476286866360917838007, −1.27890132655505043110066760710, 0, 1.27890132655505043110066760710, 2.57700579476286866360917838007, 3.30002488714173353034651158970, 4.09820171153821248929258697875, 4.73836461054798905439521732166, 5.31260740484131249293977572008, 6.31274492943501720660573990341, 7.17776444276278544734586409081, 7.82168365693021670414109571838

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