Properties

Label 2-6015-1.1-c1-0-212
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.20·2-s + 3-s + 2.85·4-s + 5-s + 2.20·6-s + 4.29·7-s + 1.87·8-s + 9-s + 2.20·10-s − 0.252·11-s + 2.85·12-s − 3.39·13-s + 9.46·14-s + 15-s − 1.56·16-s + 6.96·17-s + 2.20·18-s + 1.61·19-s + 2.85·20-s + 4.29·21-s − 0.556·22-s − 5.87·23-s + 1.87·24-s + 25-s − 7.47·26-s + 27-s + 12.2·28-s + ⋯
L(s)  = 1  + 1.55·2-s + 0.577·3-s + 1.42·4-s + 0.447·5-s + 0.899·6-s + 1.62·7-s + 0.664·8-s + 0.333·9-s + 0.696·10-s − 0.0762·11-s + 0.823·12-s − 0.940·13-s + 2.52·14-s + 0.258·15-s − 0.391·16-s + 1.68·17-s + 0.519·18-s + 0.369·19-s + 0.637·20-s + 0.937·21-s − 0.118·22-s − 1.22·23-s + 0.383·24-s + 0.200·25-s − 1.46·26-s + 0.192·27-s + 2.31·28-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: \(0\)
Selberg data: \((2,\ 6015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.179412504\)
\(L(\frac12)\) \(\approx\) \(8.179412504\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.20T + 2T^{2} \)
7 \( 1 - 4.29T + 7T^{2} \)
11 \( 1 + 0.252T + 11T^{2} \)
13 \( 1 + 3.39T + 13T^{2} \)
17 \( 1 - 6.96T + 17T^{2} \)
19 \( 1 - 1.61T + 19T^{2} \)
23 \( 1 + 5.87T + 23T^{2} \)
29 \( 1 - 3.80T + 29T^{2} \)
31 \( 1 - 9.41T + 31T^{2} \)
37 \( 1 + 7.07T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + 8.59T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 - 3.63T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 2.26T + 83T^{2} \)
89 \( 1 - 6.50T + 89T^{2} \)
97 \( 1 - 0.361T + 97T^{2} \)
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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

−7.77717292788198692931524361381, −7.50500706845457576704309039183, −6.39925602032916201172990821006, −5.70559402640563966870259751831, −4.94936473351758786238116827187, −4.67217835701708445586919094462, −3.72863357766233555009239279636, −2.89125809386093342141316531116, −2.17998586523186279394106299158, −1.31751458136120916964073232640, 1.31751458136120916964073232640, 2.17998586523186279394106299158, 2.89125809386093342141316531116, 3.72863357766233555009239279636, 4.67217835701708445586919094462, 4.94936473351758786238116827187, 5.70559402640563966870259751831, 6.39925602032916201172990821006, 7.50500706845457576704309039183, 7.77717292788198692931524361381

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