Properties

Label 2-143-1.1-c1-0-5
Degree $2$
Conductor $143$
Sign $1$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
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  + 1.23·2-s + 0.357·3-s − 0.483·4-s + 1.04·5-s + 0.439·6-s + 4.82·7-s − 3.05·8-s − 2.87·9-s + 1.28·10-s − 11-s − 0.172·12-s + 13-s + 5.94·14-s + 0.372·15-s − 2.79·16-s − 2.46·17-s − 3.53·18-s − 2.68·19-s − 0.505·20-s + 1.72·21-s − 1.23·22-s − 6.84·23-s − 1.09·24-s − 3.90·25-s + 1.23·26-s − 2.09·27-s − 2.33·28-s + ⋯
L(s)  = 1  + 0.870·2-s + 0.206·3-s − 0.241·4-s + 0.467·5-s + 0.179·6-s + 1.82·7-s − 1.08·8-s − 0.957·9-s + 0.406·10-s − 0.301·11-s − 0.0498·12-s + 0.277·13-s + 1.58·14-s + 0.0962·15-s − 0.699·16-s − 0.597·17-s − 0.833·18-s − 0.616·19-s − 0.113·20-s + 0.376·21-s − 0.262·22-s − 1.42·23-s − 0.222·24-s − 0.781·25-s + 0.241·26-s − 0.403·27-s − 0.441·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.688293950\)
\(L(\frac12)\) \(\approx\) \(1.688293950\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
13 \( 1 - T \)
good2 \( 1 - 1.23T + 2T^{2} \)
3 \( 1 - 0.357T + 3T^{2} \)
5 \( 1 - 1.04T + 5T^{2} \)
7 \( 1 - 4.82T + 7T^{2} \)
17 \( 1 + 2.46T + 17T^{2} \)
19 \( 1 + 2.68T + 19T^{2} \)
23 \( 1 + 6.84T + 23T^{2} \)
29 \( 1 - 4.25T + 29T^{2} \)
31 \( 1 - 9.87T + 31T^{2} \)
37 \( 1 - 0.450T + 37T^{2} \)
41 \( 1 + 0.139T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 3.74T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + 6.19T + 59T^{2} \)
61 \( 1 - 5.87T + 61T^{2} \)
67 \( 1 - 6.01T + 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 - 7.06T + 73T^{2} \)
79 \( 1 - 6.25T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 1.16T + 89T^{2} \)
97 \( 1 + 7.16T + 97T^{2} \)
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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

−13.61827259333092628939331305036, −12.05978409721013444605731101370, −11.43713176679695830018391218094, −10.15571583228672969549684988675, −8.624937747076016596178105140982, −8.163323367372863023764371547033, −6.17564516972940277475218634181, −5.18938021307411927282979509842, −4.18484703627831678227494085134, −2.34232815520068844557905736663, 2.34232815520068844557905736663, 4.18484703627831678227494085134, 5.18938021307411927282979509842, 6.17564516972940277475218634181, 8.163323367372863023764371547033, 8.624937747076016596178105140982, 10.15571583228672969549684988675, 11.43713176679695830018391218094, 12.05978409721013444605731101370, 13.61827259333092628939331305036

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