Properties

Label 2-1287-1.1-c1-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.53·2-s + 0.369·4-s − 3.17·5-s + 0.630·7-s + 2.51·8-s + 4.87·10-s + 11-s − 13-s − 0.971·14-s − 4.60·16-s + 0.539·17-s − 0.630·19-s − 1.17·20-s − 1.53·22-s − 1.55·23-s + 5.04·25-s + 1.53·26-s + 0.232·28-s + 8.29·29-s + 2.82·31-s + 2.06·32-s − 0.829·34-s − 2·35-s − 2.15·37-s + 0.971·38-s − 7.95·40-s + 2.63·41-s + ⋯
L(s)  = 1  − 1.08·2-s + 0.184·4-s − 1.41·5-s + 0.238·7-s + 0.887·8-s + 1.54·10-s + 0.301·11-s − 0.277·13-s − 0.259·14-s − 1.15·16-s + 0.130·17-s − 0.144·19-s − 0.261·20-s − 0.328·22-s − 0.323·23-s + 1.00·25-s + 0.301·26-s + 0.0440·28-s + 1.54·29-s + 0.508·31-s + 0.364·32-s − 0.142·34-s − 0.338·35-s − 0.354·37-s + 0.157·38-s − 1.25·40-s + 0.410·41-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: \(1\)
Selberg data: \((2,\ 1287,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53T + 2T^{2} \)
5 \( 1 + 3.17T + 5T^{2} \)
7 \( 1 - 0.630T + 7T^{2} \)
17 \( 1 - 0.539T + 17T^{2} \)
19 \( 1 + 0.630T + 19T^{2} \)
23 \( 1 + 1.55T + 23T^{2} \)
29 \( 1 - 8.29T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + 2.15T + 37T^{2} \)
41 \( 1 - 2.63T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 - 3.75T + 47T^{2} \)
53 \( 1 + 5.60T + 53T^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 + 4.18T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + 1.65T + 83T^{2} \)
89 \( 1 - 1.17T + 89T^{2} \)
97 \( 1 + 1.84T + 97T^{2} \)
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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

−9.021954932734721970528972242506, −8.517194267944697969624001534364, −7.68399835882605086235074761674, −7.32643061908013129038036256473, −6.14676676746531268529405649115, −4.65242146706143723019321022856, −4.22445578101923316079025649518, −2.91584772624314155177619790788, −1.29369002635720140723856057188, 0, 1.29369002635720140723856057188, 2.91584772624314155177619790788, 4.22445578101923316079025649518, 4.65242146706143723019321022856, 6.14676676746531268529405649115, 7.32643061908013129038036256473, 7.68399835882605086235074761674, 8.517194267944697969624001534364, 9.021954932734721970528972242506

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