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Properties

Label 2-143-11.9-c1-0-9
Degree $2$
Conductor $143$
Sign $-0.0219 + 0.999i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.690 + 2.12i)2-s + (−1 + 0.726i)3-s + (−2.42 − 1.76i)4-s + (−1 − 3.07i)5-s + (−0.854 − 2.62i)6-s + (−3.11 − 2.26i)7-s + (1.80 − 1.31i)8-s + (−0.454 + 1.40i)9-s + 7.23·10-s + (−2.80 + 1.76i)11-s + 3.70·12-s + (−0.309 + 0.951i)13-s + (6.97 − 5.06i)14-s + (3.23 + 2.35i)15-s + (−0.309 − 0.951i)16-s + (1.42 + 4.39i)17-s + ⋯
L(s)  = 1  + (−0.488 + 1.50i)2-s + (−0.577 + 0.419i)3-s + (−1.21 − 0.881i)4-s + (−0.447 − 1.37i)5-s + (−0.348 − 1.07i)6-s + (−1.17 − 0.856i)7-s + (0.639 − 0.464i)8-s + (−0.151 + 0.466i)9-s + 2.28·10-s + (−0.846 + 0.531i)11-s + 1.07·12-s + (−0.0857 + 0.263i)13-s + (1.86 − 1.35i)14-s + (0.835 + 0.607i)15-s + (−0.0772 − 0.237i)16-s + (0.346 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.0219 + 0.999i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.0219 + 0.999i)\)

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)$
bad11 \( 1 + (2.80 - 1.76i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (0.690 - 2.12i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1 - 0.726i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1 + 3.07i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.11 + 2.26i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-1.42 - 4.39i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.30 + 0.951i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (-4.54 - 3.30i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.73 + 5.34i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.23 + 2.35i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (5.11 - 3.71i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.0278 + 0.0857i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.78 - 5.65i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.26 + 6.96i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + (3.57 + 10.9i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.61 + 3.35i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.618 - 1.90i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.5 - 7.69i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + (3 - 9.23i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.02244456563924544141036695060, −12.02783708263260010778503980859, −10.35808391585397818107306726458, −9.645238451210748968974228860063, −8.351229755541325716895946892527, −7.65387794163343579744496847158, −6.32356916747566828690487028139, −5.26480915765938364689342917299, −4.25664026846416745491321257787, 0, 2.74549194341052999066172274411, 3.35705465812046926959906764985, 5.86147924833774241363835686485, 6.88104661188039067235436767130, 8.402204113829066560864920809739, 9.806519314156690840102396892248, 10.33454093666401445333007066333, 11.59825093230925005584423692050, 11.90969529156266189687814681559

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