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Properties

Label 2-143-13.3-c1-0-9
Degree $2$
Conductor $143$
Sign $-0.662 + 0.749i$
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  + (−0.5 − 0.866i)2-s + (−1.30 − 2.25i)3-s + (0.500 − 0.866i)4-s + 3.76·5-s + (−1.30 + 2.25i)6-s + (−0.0835 + 0.144i)7-s − 3·8-s + (−1.88 + 3.26i)9-s + (−1.88 − 3.26i)10-s + (−0.5 − 0.866i)11-s − 2.60·12-s + (0.199 + 3.60i)13-s + 0.167·14-s + (−4.90 − 8.49i)15-s + (0.500 + 0.866i)16-s + (−2.18 + 3.78i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.751 − 1.30i)3-s + (0.250 − 0.433i)4-s + 1.68·5-s + (−0.531 + 0.919i)6-s + (−0.0315 + 0.0546i)7-s − 1.06·8-s + (−0.628 + 1.08i)9-s + (−0.595 − 1.03i)10-s + (−0.150 − 0.261i)11-s − 0.751·12-s + (0.0552 + 0.998i)13-s + 0.0446·14-s + (−1.26 − 2.19i)15-s + (0.125 + 0.216i)16-s + (−0.529 + 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.662 + 0.749i$
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: \((2,\ 143,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.393405 - 0.872638i\)
\(L(\frac12)\) \(\approx\) \(0.393405 - 0.872638i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.199 - 3.60i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.30 + 2.25i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.76T + 5T^{2} \)
7 \( 1 + (0.0835 - 0.144i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.68 + 2.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.60 - 4.50i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.967 + 1.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.86T + 31T^{2} \)
37 \( 1 + (4.31 + 7.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.633 - 1.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.60 + 6.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 1.33T + 53T^{2} \)
59 \( 1 + (1.06 - 1.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.40 + 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.06 - 3.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (7.15 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.78 - 3.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.72826992280505370952037309872, −11.62569672545782006515828101109, −10.87246515019325230800700496814, −9.783009441604310892954384095437, −8.876701740028882947880482124057, −6.96642708192408383647240671901, −6.21302881003419412178982298768, −5.45577233419780419919168320201, −2.33448561716203863438677919592, −1.38335798203332353205906776523, 2.91724849921526234515690585624, 4.90221327232450364826769034620, 5.81053370930250846943827634553, 6.77831233247347551298271155952, 8.453062993089041897617818191866, 9.615600084736281348359636081597, 10.13568424336519535674425207219, 11.18654381735599218073917420652, 12.44717552100503900270911124591, 13.53990537373612507181701169113

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