Properties

Label 2-1143-1.1-c1-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.06·2-s + 2.25·4-s − 3.73·5-s + 1.84·7-s − 0.536·8-s + 7.70·10-s + 4.36·11-s − 5.39·13-s − 3.80·14-s − 3.41·16-s − 4.13·17-s + 0.483·19-s − 8.43·20-s − 9.01·22-s + 2.47·23-s + 8.93·25-s + 11.1·26-s + 4.16·28-s + 8.03·29-s + 0.894·31-s + 8.11·32-s + 8.53·34-s − 6.88·35-s + 7.79·37-s − 0.997·38-s + 2.00·40-s − 1.77·41-s + ⋯
L(s)  = 1  − 1.45·2-s + 1.12·4-s − 1.66·5-s + 0.696·7-s − 0.189·8-s + 2.43·10-s + 1.31·11-s − 1.49·13-s − 1.01·14-s − 0.853·16-s − 1.00·17-s + 0.110·19-s − 1.88·20-s − 1.92·22-s + 0.515·23-s + 1.78·25-s + 2.18·26-s + 0.787·28-s + 1.49·29-s + 0.160·31-s + 1.43·32-s + 1.46·34-s − 1.16·35-s + 1.28·37-s − 0.161·38-s + 0.316·40-s − 0.277·41-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: \(1\)
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 \( 1 + 2.06T + 2T^{2} \)
5 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 - 1.84T + 7T^{2} \)
11 \( 1 - 4.36T + 11T^{2} \)
13 \( 1 + 5.39T + 13T^{2} \)
17 \( 1 + 4.13T + 17T^{2} \)
19 \( 1 - 0.483T + 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 - 8.03T + 29T^{2} \)
31 \( 1 - 0.894T + 31T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + 1.77T + 41T^{2} \)
43 \( 1 + 1.03T + 43T^{2} \)
47 \( 1 + 8.87T + 47T^{2} \)
53 \( 1 + 4.17T + 53T^{2} \)
59 \( 1 + 0.500T + 59T^{2} \)
61 \( 1 - 7.90T + 61T^{2} \)
67 \( 1 + 8.39T + 67T^{2} \)
71 \( 1 + 7.26T + 71T^{2} \)
73 \( 1 + 5.37T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 - 0.247T + 89T^{2} \)
97 \( 1 + 5.42T + 97T^{2} \)
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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

−9.251354134319029355503198794862, −8.486634510222635608051571806390, −7.978694448020179450539183622301, −7.16958053400496203862024648827, −6.65904652323652706694366205904, −4.70343733553419295692001625380, −4.31682396594333930306249077955, −2.80310813857017585226562085145, −1.31988267610331545396315701572, 0, 1.31988267610331545396315701572, 2.80310813857017585226562085145, 4.31682396594333930306249077955, 4.70343733553419295692001625380, 6.65904652323652706694366205904, 7.16958053400496203862024648827, 7.978694448020179450539183622301, 8.486634510222635608051571806390, 9.251354134319029355503198794862

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