Properties

Label 2-1287-1.1-c1-0-15
Degree $2$
Conductor $1287$
Sign $1$
Analytic cond. $10.2767$
Root an. cond. $3.20573$
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  + 1.79·2-s + 1.23·4-s − 3.97·5-s + 3.17·7-s − 1.37·8-s − 7.14·10-s + 11-s + 13-s + 5.70·14-s − 4.94·16-s + 7.80·17-s + 7.17·19-s − 4.90·20-s + 1.79·22-s − 3.36·23-s + 10.7·25-s + 1.79·26-s + 3.92·28-s + 7.61·29-s + 3.15·31-s − 6.14·32-s + 14.0·34-s − 12.6·35-s + 2.93·37-s + 12.9·38-s + 5.45·40-s − 1.64·41-s + ⋯
L(s)  = 1  + 1.27·2-s + 0.617·4-s − 1.77·5-s + 1.19·7-s − 0.485·8-s − 2.25·10-s + 0.301·11-s + 0.277·13-s + 1.52·14-s − 1.23·16-s + 1.89·17-s + 1.64·19-s − 1.09·20-s + 0.383·22-s − 0.700·23-s + 2.15·25-s + 0.352·26-s + 0.741·28-s + 1.41·29-s + 0.566·31-s − 1.08·32-s + 2.40·34-s − 2.13·35-s + 0.482·37-s + 2.09·38-s + 0.863·40-s − 0.256·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(10.2767\)
Root analytic conductor: \(3.20573\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.675539767\)
\(L(\frac12)\) \(\approx\) \(2.675539767\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 - 1.79T + 2T^{2} \)
5 \( 1 + 3.97T + 5T^{2} \)
7 \( 1 - 3.17T + 7T^{2} \)
17 \( 1 - 7.80T + 17T^{2} \)
19 \( 1 - 7.17T + 19T^{2} \)
23 \( 1 + 3.36T + 23T^{2} \)
29 \( 1 - 7.61T + 29T^{2} \)
31 \( 1 - 3.15T + 31T^{2} \)
37 \( 1 - 2.93T + 37T^{2} \)
41 \( 1 + 1.64T + 41T^{2} \)
43 \( 1 + 4.15T + 43T^{2} \)
47 \( 1 + 0.660T + 47T^{2} \)
53 \( 1 + 0.0696T + 53T^{2} \)
59 \( 1 + 8.88T + 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 - 5.50T + 67T^{2} \)
71 \( 1 + 2.11T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 0.965T + 89T^{2} \)
97 \( 1 - 7.31T + 97T^{2} \)
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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

−9.717596359054852970535207594608, −8.497947114900523271008348269507, −7.923064271632846373200274015162, −7.32127763598323915871527713842, −6.11553225555556186191053193310, −5.07273721552682611527301502254, −4.58336187211376835845511904352, −3.62848503959207029877219619621, −3.07753456728500260131097729389, −1.07835797644668383411200575987, 1.07835797644668383411200575987, 3.07753456728500260131097729389, 3.62848503959207029877219619621, 4.58336187211376835845511904352, 5.07273721552682611527301502254, 6.11553225555556186191053193310, 7.32127763598323915871527713842, 7.923064271632846373200274015162, 8.497947114900523271008348269507, 9.717596359054852970535207594608

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