Properties

Label 2-6045-1.1-c1-0-22
Degree $2$
Conductor $6045$
Sign $1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
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  + 1.26·2-s − 3-s − 0.392·4-s − 5-s − 1.26·6-s − 1.11·7-s − 3.03·8-s + 9-s − 1.26·10-s − 0.675·11-s + 0.392·12-s − 13-s − 1.41·14-s + 15-s − 3.06·16-s + 6.69·17-s + 1.26·18-s − 4.85·19-s + 0.392·20-s + 1.11·21-s − 0.856·22-s − 0.453·23-s + 3.03·24-s + 25-s − 1.26·26-s − 27-s + 0.439·28-s + ⋯
L(s)  = 1  + 0.896·2-s − 0.577·3-s − 0.196·4-s − 0.447·5-s − 0.517·6-s − 0.423·7-s − 1.07·8-s + 0.333·9-s − 0.400·10-s − 0.203·11-s + 0.113·12-s − 0.277·13-s − 0.379·14-s + 0.258·15-s − 0.765·16-s + 1.62·17-s + 0.298·18-s − 1.11·19-s + 0.0877·20-s + 0.244·21-s − 0.182·22-s − 0.0945·23-s + 0.619·24-s + 0.200·25-s − 0.248·26-s − 0.192·27-s + 0.0830·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(48.2695\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069242036\)
\(L(\frac12)\) \(\approx\) \(1.069242036\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
31 \( 1 - T \)
good2 \( 1 - 1.26T + 2T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 + 0.675T + 11T^{2} \)
17 \( 1 - 6.69T + 17T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
23 \( 1 + 0.453T + 23T^{2} \)
29 \( 1 + 5.46T + 29T^{2} \)
37 \( 1 + 9.92T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + 7.54T + 47T^{2} \)
53 \( 1 - 8.73T + 53T^{2} \)
59 \( 1 - 0.468T + 59T^{2} \)
61 \( 1 - 9.96T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + 8.91T + 79T^{2} \)
83 \( 1 - 9.09T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−8.090997889699375377001355108046, −7.12292785125187658602781719071, −6.54490047525174108091499345250, −5.68450114267846477627181162764, −5.24227805148078577004697609499, −4.51677416244750009656013211172, −3.62559736055102433189793756932, −3.25461451013677116654588208123, −1.95225423306187657221571684885, −0.47259469626452903222535558594, 0.47259469626452903222535558594, 1.95225423306187657221571684885, 3.25461451013677116654588208123, 3.62559736055102433189793756932, 4.51677416244750009656013211172, 5.24227805148078577004697609499, 5.68450114267846477627181162764, 6.54490047525174108091499345250, 7.12292785125187658602781719071, 8.090997889699375377001355108046

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