Properties

Label 2-1143-1.1-c1-0-49
Degree $2$
Conductor $1143$
Sign $1$
Analytic cond. $9.12690$
Root an. cond. $3.02107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 3·5-s − 4·7-s + 6·10-s − 6·11-s − 7·13-s + 8·14-s − 4·16-s + 2·17-s − 6·20-s + 12·22-s − 23-s + 4·25-s + 14·26-s − 8·28-s − 9·29-s − 5·31-s + 8·32-s − 4·34-s + 12·35-s − 3·37-s + 6·41-s + 4·43-s − 12·44-s + 2·46-s − 2·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.34·5-s − 1.51·7-s + 1.89·10-s − 1.80·11-s − 1.94·13-s + 2.13·14-s − 16-s + 0.485·17-s − 1.34·20-s + 2.55·22-s − 0.208·23-s + 4/5·25-s + 2.74·26-s − 1.51·28-s − 1.67·29-s − 0.898·31-s + 1.41·32-s − 0.685·34-s + 2.02·35-s − 0.493·37-s + 0.937·41-s + 0.609·43-s − 1.80·44-s + 0.294·46-s − 0.291·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1143\)    =    \(3^{2} \cdot 127\)
Sign: $1$
Analytic conductor: \(9.12690\)
Root analytic conductor: \(3.02107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 1143,\ (\ :1/2),\ 1)\)

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 \( 1 \)
127 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.173685595792635956288106677373, −7.979069184297479970157387427743, −7.50237058980749934967978419558, −7.14586714404397132780454647934, −5.67487786880756741852024245547, −4.56876609912340916029501181301, −3.31695645330805523753956502481, −2.38337836408984736978763024018, 0, 0, 2.38337836408984736978763024018, 3.31695645330805523753956502481, 4.56876609912340916029501181301, 5.67487786880756741852024245547, 7.14586714404397132780454647934, 7.50237058980749934967978419558, 7.979069184297479970157387427743, 9.173685595792635956288106677373

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