Properties

Label 2-143-13.3-c1-0-10
Degree $2$
Conductor $143$
Sign $-0.894 + 0.447i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.134 − 0.232i)2-s + (−1.17 − 2.03i)3-s + (0.963 − 1.66i)4-s − 2.80·5-s + (−0.315 + 0.546i)6-s + (−1.19 + 2.06i)7-s − 1.05·8-s + (−1.26 + 2.18i)9-s + (0.376 + 0.652i)10-s + (0.5 + 0.866i)11-s − 4.53·12-s + (1.15 − 3.41i)13-s + 0.640·14-s + (3.30 + 5.71i)15-s + (−1.78 − 3.09i)16-s + (0.918 − 1.59i)17-s + ⋯
L(s)  = 1  + (−0.0948 − 0.164i)2-s + (−0.678 − 1.17i)3-s + (0.481 − 0.834i)4-s − 1.25·5-s + (−0.128 + 0.223i)6-s + (−0.450 + 0.780i)7-s − 0.372·8-s + (−0.421 + 0.729i)9-s + (0.119 + 0.206i)10-s + (0.150 + 0.261i)11-s − 1.30·12-s + (0.321 − 0.946i)13-s + 0.171·14-s + (0.852 + 1.47i)15-s + (−0.446 − 0.773i)16-s + (0.222 − 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147030 - 0.621743i\)
\(L(\frac12)\) \(\approx\) \(0.147030 - 0.621743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-1.15 + 3.41i)T \)
good2 \( 1 + (0.134 + 0.232i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.17 + 2.03i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.80T + 5T^{2} \)
7 \( 1 + (1.19 - 2.06i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-0.918 + 1.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.00 + 6.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.83 + 4.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.35 - 4.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.36T + 31T^{2} \)
37 \( 1 + (-4.85 - 8.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.76 - 3.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.709 + 1.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.28T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + (-4.36 + 7.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.96 + 6.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.331 - 0.574i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.77 - 3.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.81T + 73T^{2} \)
79 \( 1 - 9.63T + 79T^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 + (1.53 + 2.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.765 + 1.32i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.34689222848073787761043614057, −11.76764100104814312403205899061, −11.07029706431827969421799141690, −9.709090540615833943756142304814, −8.257867135055330431499405918330, −7.10255490934967495823214399440, −6.32029452658251004290400390541, −5.09834950296201905232082803784, −2.81836438520194874214522193447, −0.72707416313253218939260345193, 3.83219945796032975779583328898, 3.92810172925826137548109869614, 5.91829629617562686955733909102, 7.29268685658349408449470130112, 8.143705413260170694665251420173, 9.590111935187345438208366356354, 10.63344690733222736314384754204, 11.65932018443111123393796405115, 11.97829890129588502196396607000, 13.50300065336982582924674330463

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