Properties

Label 2-1143-1.1-c1-0-26
Degree $2$
Conductor $1143$
Sign $1$
Analytic cond. $9.12690$
Root an. cond. $3.02107$
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  − 1.71·2-s + 0.936·4-s + 4.13·5-s + 0.967·7-s + 1.82·8-s − 7.08·10-s + 6.54·11-s + 2.28·13-s − 1.65·14-s − 4.99·16-s + 2.37·17-s + 2.94·19-s + 3.87·20-s − 11.2·22-s + 1.52·23-s + 12.0·25-s − 3.91·26-s + 0.906·28-s − 2.59·29-s − 4.67·31-s + 4.91·32-s − 4.07·34-s + 4·35-s − 9.81·37-s − 5.03·38-s + 7.53·40-s − 4.37·41-s + ⋯
L(s)  = 1  − 1.21·2-s + 0.468·4-s + 1.84·5-s + 0.365·7-s + 0.644·8-s − 2.24·10-s + 1.97·11-s + 0.632·13-s − 0.443·14-s − 1.24·16-s + 0.576·17-s + 0.674·19-s + 0.866·20-s − 2.39·22-s + 0.317·23-s + 2.41·25-s − 0.767·26-s + 0.171·28-s − 0.481·29-s − 0.840·31-s + 0.869·32-s − 0.698·34-s + 0.676·35-s − 1.61·37-s − 0.817·38-s + 1.19·40-s − 0.682·41-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1143,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.468586721\)
\(L(\frac12)\) \(\approx\) \(1.468586721\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.71T + 2T^{2} \)
5 \( 1 - 4.13T + 5T^{2} \)
7 \( 1 - 0.967T + 7T^{2} \)
11 \( 1 - 6.54T + 11T^{2} \)
13 \( 1 - 2.28T + 13T^{2} \)
17 \( 1 - 2.37T + 17T^{2} \)
19 \( 1 - 2.94T + 19T^{2} \)
23 \( 1 - 1.52T + 23T^{2} \)
29 \( 1 + 2.59T + 29T^{2} \)
31 \( 1 + 4.67T + 31T^{2} \)
37 \( 1 + 9.81T + 37T^{2} \)
41 \( 1 + 4.37T + 41T^{2} \)
43 \( 1 - 1.55T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 3.66T + 59T^{2} \)
61 \( 1 + 0.791T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + 6.43T + 71T^{2} \)
73 \( 1 + 3.78T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 - 3.28T + 83T^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
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

−9.688071566202146880429660846752, −9.007568843498402267738629403823, −8.672796452244273995463424978001, −7.34007248964104204904397808289, −6.58733657214333798063424164442, −5.77790215072254009820905041215, −4.79496503759200843834659750692, −3.41031719193765221419338716851, −1.65840649958548290235169214077, −1.40851084851587041099492579796, 1.40851084851587041099492579796, 1.65840649958548290235169214077, 3.41031719193765221419338716851, 4.79496503759200843834659750692, 5.77790215072254009820905041215, 6.58733657214333798063424164442, 7.34007248964104204904397808289, 8.672796452244273995463424978001, 9.007568843498402267738629403823, 9.688071566202146880429660846752

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