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Properties

Label 2-143-13.3-c1-0-11
Degree $2$
Conductor $143$
Sign $0.0128 - 0.999i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 − 2.17i)2-s + (−1.14 − 1.97i)3-s + (−2.14 + 3.71i)4-s − 1.22·5-s + (−2.86 + 4.96i)6-s + (−0.389 + 0.673i)7-s + 5.72·8-s + (−1.11 + 1.92i)9-s + (1.53 + 2.65i)10-s + (−0.5 − 0.866i)11-s + 9.79·12-s + (−2.5 − 2.59i)13-s + 1.95·14-s + (1.39 + 2.41i)15-s + (−2.89 − 5.01i)16-s + (2.14 − 3.71i)17-s + ⋯
L(s)  = 1  + (−0.886 − 1.53i)2-s + (−0.659 − 1.14i)3-s + (−1.07 + 1.85i)4-s − 0.546·5-s + (−1.16 + 2.02i)6-s + (−0.147 + 0.254i)7-s + 2.02·8-s + (−0.370 + 0.641i)9-s + (0.484 + 0.838i)10-s + (−0.150 − 0.261i)11-s + 2.82·12-s + (−0.693 − 0.720i)13-s + 0.521·14-s + (0.360 + 0.624i)15-s + (−0.724 − 1.25i)16-s + (0.519 − 0.900i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.0128 - 0.999i$
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: \((2,\ 143,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159218 + 0.157189i\)
\(L(\frac12)\) \(\approx\) \(0.159218 + 0.157189i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (1.25 + 2.17i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.14 + 1.97i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.22T + 5T^{2} \)
7 \( 1 + (0.389 - 0.673i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-2.14 + 3.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.14 - 7.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.36 - 2.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.68 + 6.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.44T + 31T^{2} \)
37 \( 1 + (-0.396 - 0.686i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.87 + 8.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.38T + 47T^{2} \)
53 \( 1 - 1.79T + 53T^{2} \)
59 \( 1 + (2.60 - 4.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.26 + 12.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.03 + 5.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.25 + 5.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + (-2.38 - 4.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.07 + 3.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.19549008968681254020060031288, −11.54328833674430750170517052766, −10.49876065132738315627611416595, −9.503965398411223857990635321031, −8.114723370273521803292256954108, −7.45423016647127377876987861942, −5.72732559575342825455081354380, −3.65929259269781138913792962529, −2.06327866109594052857626405340, −0.31571882517742124963232595607, 4.28452767100056578507054895757, 5.18891349754117459711667445904, 6.49918513508715073043005549322, 7.46095784450885393330740664438, 8.690799655618105459951804704769, 9.622935173305187678681025156822, 10.45398444003953861089555367642, 11.43246446199094353214870217953, 13.03625647269013510250945280421, 14.64291672343852923368178856228

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