Properties

Label 2-1287-1.1-c1-0-17
Degree $2$
Conductor $1287$
Sign $1$
Analytic cond. $10.2767$
Root an. cond. $3.20573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s − 3·8-s + 2·10-s + 11-s + 13-s − 16-s + 6·17-s − 4·19-s − 2·20-s + 22-s + 8·23-s − 25-s + 26-s + 10·29-s + 5·32-s + 6·34-s + 6·37-s − 4·38-s − 6·40-s − 10·41-s + 4·43-s − 44-s + 8·46-s − 8·47-s − 7·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s + 0.301·11-s + 0.277·13-s − 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.447·20-s + 0.213·22-s + 1.66·23-s − 1/5·25-s + 0.196·26-s + 1.85·29-s + 0.883·32-s + 1.02·34-s + 0.986·37-s − 0.648·38-s − 0.948·40-s − 1.56·41-s + 0.609·43-s − 0.150·44-s + 1.17·46-s − 1.16·47-s − 49-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.422614242\)
\(L(\frac12)\) \(\approx\) \(2.422614242\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.828201966522165801644231083131, −8.809781988911382660911278540879, −8.279842457480330856862908947423, −6.91274116401925090743914373893, −6.14885963657490762630313444248, −5.39216574159615869785069323377, −4.65912153519243395995881297600, −3.58566955557413367386868416593, −2.66792787949502047211352973601, −1.12169940393228089488805712125, 1.12169940393228089488805712125, 2.66792787949502047211352973601, 3.58566955557413367386868416593, 4.65912153519243395995881297600, 5.39216574159615869785069323377, 6.14885963657490762630313444248, 6.91274116401925090743914373893, 8.279842457480330856862908947423, 8.809781988911382660911278540879, 9.828201966522165801644231083131

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