Properties

Label 2-1143-1.1-c1-0-32
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  + 2.55·2-s + 4.52·4-s − 2.44·5-s + 3.89·7-s + 6.43·8-s − 6.25·10-s − 3.82·11-s + 4.41·13-s + 9.95·14-s + 7.39·16-s + 7.77·17-s − 5.77·19-s − 11.0·20-s − 9.76·22-s + 4.71·23-s + 0.997·25-s + 11.2·26-s + 17.6·28-s + 1.05·29-s − 0.879·31-s + 6.00·32-s + 19.8·34-s − 9.54·35-s + 2.79·37-s − 14.7·38-s − 15.7·40-s − 0.154·41-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.26·4-s − 1.09·5-s + 1.47·7-s + 2.27·8-s − 1.97·10-s − 1.15·11-s + 1.22·13-s + 2.65·14-s + 1.84·16-s + 1.88·17-s − 1.32·19-s − 2.47·20-s − 2.08·22-s + 0.983·23-s + 0.199·25-s + 2.21·26-s + 3.32·28-s + 0.194·29-s − 0.157·31-s + 1.06·32-s + 3.40·34-s − 1.61·35-s + 0.460·37-s − 2.39·38-s − 2.49·40-s − 0.0240·41-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

\(L(1)\) \(\approx\) \(4.705606788\)
\(L(\frac12)\) \(\approx\) \(4.705606788\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.55T + 2T^{2} \)
5 \( 1 + 2.44T + 5T^{2} \)
7 \( 1 - 3.89T + 7T^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 - 7.77T + 17T^{2} \)
19 \( 1 + 5.77T + 19T^{2} \)
23 \( 1 - 4.71T + 23T^{2} \)
29 \( 1 - 1.05T + 29T^{2} \)
31 \( 1 + 0.879T + 31T^{2} \)
37 \( 1 - 2.79T + 37T^{2} \)
41 \( 1 + 0.154T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 - 5.76T + 53T^{2} \)
59 \( 1 + 4.10T + 59T^{2} \)
61 \( 1 - 4.94T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 6.49T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + 2.55T + 89T^{2} \)
97 \( 1 + 2.23T + 97T^{2} \)
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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

−10.43807392157205619423842538112, −8.507028679152427391419358099351, −7.947400391496155964048428778464, −7.29419443165245705289157396095, −6.11286497296409233647778949190, −5.25855839693860939189732734282, −4.64598153347081893140279843581, −3.77979210783086930726757326662, −2.96435917465414854015847274144, −1.56283748009289920414005076144, 1.56283748009289920414005076144, 2.96435917465414854015847274144, 3.77979210783086930726757326662, 4.64598153347081893140279843581, 5.25855839693860939189732734282, 6.11286497296409233647778949190, 7.29419443165245705289157396095, 7.947400391496155964048428778464, 8.507028679152427391419358099351, 10.43807392157205619423842538112

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