Properties

Label 2-285-1.1-c1-0-7
Degree $2$
Conductor $285$
Sign $1$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + 2.64·2-s − 3-s + 5.00·4-s + 5-s − 2.64·6-s − 3.64·7-s + 7.93·8-s + 9-s + 2.64·10-s + 5.64·11-s − 5.00·12-s − 5.64·13-s − 9.64·14-s − 15-s + 11.0·16-s − 4·17-s + 2.64·18-s − 19-s + 5.00·20-s + 3.64·21-s + 14.9·22-s − 1.29·23-s − 7.93·24-s + 25-s − 14.9·26-s − 27-s − 18.2·28-s + ⋯
L(s)  = 1  + 1.87·2-s − 0.577·3-s + 2.50·4-s + 0.447·5-s − 1.08·6-s − 1.37·7-s + 2.80·8-s + 0.333·9-s + 0.836·10-s + 1.70·11-s − 1.44·12-s − 1.56·13-s − 2.57·14-s − 0.258·15-s + 2.75·16-s − 0.970·17-s + 0.623·18-s − 0.229·19-s + 1.11·20-s + 0.795·21-s + 3.18·22-s − 0.269·23-s − 1.62·24-s + 0.200·25-s − 2.92·26-s − 0.192·27-s − 3.44·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.989732226\)
\(L(\frac12)\) \(\approx\) \(2.989732226\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 2.64T + 2T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 + 5.64T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
23 \( 1 + 1.29T + 23T^{2} \)
29 \( 1 + 6.93T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 1.64T + 37T^{2} \)
41 \( 1 + 4.35T + 41T^{2} \)
43 \( 1 - 0.354T + 43T^{2} \)
47 \( 1 - 9.29T + 47T^{2} \)
53 \( 1 - 0.708T + 53T^{2} \)
59 \( 1 - 0.708T + 59T^{2} \)
61 \( 1 + 0.708T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 1.06T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−12.13837975172374549503100235117, −11.38756509348376275999744915614, −10.18739812158716040263250652269, −9.308506745153719777501357755781, −7.07298766388026975824108747754, −6.59347835013690321556803685122, −5.79765047757978808166933689828, −4.61530551040323117217267653120, −3.65404537443948603368575444572, −2.26979476609353166891886826979, 2.26979476609353166891886826979, 3.65404537443948603368575444572, 4.61530551040323117217267653120, 5.79765047757978808166933689828, 6.59347835013690321556803685122, 7.07298766388026975824108747754, 9.308506745153719777501357755781, 10.18739812158716040263250652269, 11.38756509348376275999744915614, 12.13837975172374549503100235117

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