Properties

Label 2-1287-1.1-c1-0-18
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  − 0.178·2-s − 1.96·4-s + 2.75·5-s + 3.42·7-s + 0.707·8-s − 0.490·10-s − 11-s + 13-s − 0.611·14-s + 3.81·16-s + 2.82·17-s − 1.24·19-s − 5.41·20-s + 0.178·22-s + 3.70·23-s + 2.57·25-s − 0.178·26-s − 6.74·28-s − 2.18·29-s − 0.285·31-s − 2.09·32-s − 0.503·34-s + 9.43·35-s − 6.16·37-s + 0.221·38-s + 1.94·40-s + 5.67·41-s + ⋯
L(s)  = 1  − 0.126·2-s − 0.984·4-s + 1.23·5-s + 1.29·7-s + 0.250·8-s − 0.155·10-s − 0.301·11-s + 0.277·13-s − 0.163·14-s + 0.952·16-s + 0.684·17-s − 0.285·19-s − 1.21·20-s + 0.0380·22-s + 0.772·23-s + 0.514·25-s − 0.0349·26-s − 1.27·28-s − 0.404·29-s − 0.0513·31-s − 0.370·32-s − 0.0863·34-s + 1.59·35-s − 1.01·37-s + 0.0359·38-s + 0.307·40-s + 0.886·41-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

\(L(1)\) \(\approx\) \(1.848135243\)
\(L(\frac12)\) \(\approx\) \(1.848135243\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.178T + 2T^{2} \)
5 \( 1 - 2.75T + 5T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
23 \( 1 - 3.70T + 23T^{2} \)
29 \( 1 + 2.18T + 29T^{2} \)
31 \( 1 + 0.285T + 31T^{2} \)
37 \( 1 + 6.16T + 37T^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 + 1.42T + 43T^{2} \)
47 \( 1 - 2.68T + 47T^{2} \)
53 \( 1 - 7.75T + 53T^{2} \)
59 \( 1 + 4.12T + 59T^{2} \)
61 \( 1 + 3.85T + 61T^{2} \)
67 \( 1 + 8.93T + 67T^{2} \)
71 \( 1 - 16.4T + 71T^{2} \)
73 \( 1 - 7.00T + 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 - 18.1T + 83T^{2} \)
89 \( 1 - 8.71T + 89T^{2} \)
97 \( 1 - 0.384T + 97T^{2} \)
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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

−9.544365436762793103434944387301, −8.939796859584226573214192529208, −8.174765839204499282023254425750, −7.41932276638608354069881232779, −6.11215590174570274432819771209, −5.30597803357904821874068296311, −4.81591829739708304183417136809, −3.62531301753966563141677945084, −2.16622770836686093860526138631, −1.13613332692143935640802598006, 1.13613332692143935640802598006, 2.16622770836686093860526138631, 3.62531301753966563141677945084, 4.81591829739708304183417136809, 5.30597803357904821874068296311, 6.11215590174570274432819771209, 7.41932276638608354069881232779, 8.174765839204499282023254425750, 8.939796859584226573214192529208, 9.544365436762793103434944387301

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