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Properties

Label 2-143-13.9-c1-0-5
Degree $2$
Conductor $143$
Sign $0.942 - 0.333i$
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  + (0.910 − 1.57i)2-s + (−1.55 + 2.69i)3-s + (−0.659 − 1.14i)4-s + 0.0854·5-s + (2.83 + 4.91i)6-s + (2.25 + 3.91i)7-s + 1.24·8-s + (−3.35 − 5.80i)9-s + (0.0778 − 0.134i)10-s + (0.5 − 0.866i)11-s + 4.10·12-s + (−3.26 − 1.53i)13-s + 8.23·14-s + (−0.133 + 0.230i)15-s + (2.44 − 4.24i)16-s + (2.04 + 3.54i)17-s + ⋯
L(s)  = 1  + (0.644 − 1.11i)2-s + (−0.899 + 1.55i)3-s + (−0.329 − 0.570i)4-s + 0.0382·5-s + (1.15 + 2.00i)6-s + (0.854 + 1.47i)7-s + 0.438·8-s + (−1.11 − 1.93i)9-s + (0.0246 − 0.0426i)10-s + (0.150 − 0.261i)11-s + 1.18·12-s + (−0.905 − 0.424i)13-s + 2.20·14-s + (−0.0343 + 0.0595i)15-s + (0.612 − 1.06i)16-s + (0.496 + 0.859i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24049 + 0.212960i\)
\(L(\frac12)\) \(\approx\) \(1.24049 + 0.212960i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.26 + 1.53i)T \)
good2 \( 1 + (-0.910 + 1.57i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.55 - 2.69i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.0854T + 5T^{2} \)
7 \( 1 + (-2.25 - 3.91i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-2.04 - 3.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.33 + 2.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.20 + 2.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.37 + 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.664T + 31T^{2} \)
37 \( 1 + (-0.467 + 0.810i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.903 + 1.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.484 - 0.839i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.118T + 47T^{2} \)
53 \( 1 - 2.80T + 53T^{2} \)
59 \( 1 + (1.58 + 2.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.108 - 0.187i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.75 + 6.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.27 - 12.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.37T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 + 8.24T + 83T^{2} \)
89 \( 1 + (5.02 - 8.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.203 + 0.353i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.63726333662305487073095861040, −11.86051091730073207068245594770, −11.34738795523027517985693262427, −10.39097853559500832886182307049, −9.573841209287356187046719738253, −8.278940556777056940603914255530, −5.87394787251381543683983224756, −5.05854754074035727830655635709, −4.12350825122778399163373494411, −2.59956127322273658982498386902, 1.45849150752071123007486358881, 4.54766776735773588374725137853, 5.51499624694326737759856669269, 6.82102632612914072745589518581, 7.29098196130367918629098407667, 7.996373775909904017880377609727, 10.27681636902939547623254739220, 11.34077863248559941931109442916, 12.27347451222994534776964894269, 13.32977320516998066074908404846

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