Properties

Label 2-1287-1.1-c1-0-16
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  − 2.57·2-s + 4.63·4-s + 4.12·5-s + 2.15·7-s − 6.79·8-s − 10.6·10-s − 11-s − 13-s − 5.56·14-s + 8.23·16-s + 3.09·17-s − 2.68·19-s + 19.1·20-s + 2.57·22-s − 3.31·23-s + 11.9·25-s + 2.57·26-s + 10.0·28-s + 3.34·29-s + 9.96·31-s − 7.62·32-s − 7.96·34-s + 8.89·35-s + 2.05·37-s + 6.92·38-s − 28.0·40-s − 9.49·41-s + ⋯
L(s)  = 1  − 1.82·2-s + 2.31·4-s + 1.84·5-s + 0.815·7-s − 2.40·8-s − 3.35·10-s − 0.301·11-s − 0.277·13-s − 1.48·14-s + 2.05·16-s + 0.750·17-s − 0.616·19-s + 4.27·20-s + 0.549·22-s − 0.690·23-s + 2.39·25-s + 0.505·26-s + 1.89·28-s + 0.621·29-s + 1.79·31-s − 1.34·32-s − 1.36·34-s + 1.50·35-s + 0.338·37-s + 1.12·38-s − 4.43·40-s − 1.48·41-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

\(L(1)\) \(\approx\) \(1.133658556\)
\(L(\frac12)\) \(\approx\) \(1.133658556\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.57T + 2T^{2} \)
5 \( 1 - 4.12T + 5T^{2} \)
7 \( 1 - 2.15T + 7T^{2} \)
17 \( 1 - 3.09T + 17T^{2} \)
19 \( 1 + 2.68T + 19T^{2} \)
23 \( 1 + 3.31T + 23T^{2} \)
29 \( 1 - 3.34T + 29T^{2} \)
31 \( 1 - 9.96T + 31T^{2} \)
37 \( 1 - 2.05T + 37T^{2} \)
41 \( 1 + 9.49T + 41T^{2} \)
43 \( 1 - 2.84T + 43T^{2} \)
47 \( 1 - 9.46T + 47T^{2} \)
53 \( 1 - 5.34T + 53T^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 + 0.193T + 61T^{2} \)
67 \( 1 + 5.78T + 67T^{2} \)
71 \( 1 - 3.78T + 71T^{2} \)
73 \( 1 - 7.47T + 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 - 0.218T + 83T^{2} \)
89 \( 1 - 1.52T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−9.829589878703468912635445953848, −8.847190248880387292819016869996, −8.326281159581839268810360653705, −7.45424062007376343066592363039, −6.47823639452654207318615190627, −5.87295501180954564273830504977, −4.82708222999894990652721794750, −2.73782503843123856590455694189, −2.01036214049247297134011944857, −1.10296127109661948844454050543, 1.10296127109661948844454050543, 2.01036214049247297134011944857, 2.73782503843123856590455694189, 4.82708222999894990652721794750, 5.87295501180954564273830504977, 6.47823639452654207318615190627, 7.45424062007376343066592363039, 8.326281159581839268810360653705, 8.847190248880387292819016869996, 9.829589878703468912635445953848

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