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Properties

Label 2-143-13.8-c2-0-12
Degree $2$
Conductor $143$
Sign $0.670 - 0.741i$
Analytic cond. $3.89646$
Root an. cond. $1.97394$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.53 + 1.53i)2-s + 1.81·3-s + 0.728i·4-s + (3.39 + 3.39i)5-s + (2.79 + 2.79i)6-s + (4.13 − 4.13i)7-s + (5.03 − 5.03i)8-s − 5.70·9-s + 10.4i·10-s + (−2.34 + 2.34i)11-s + 1.32i·12-s + (−12.2 + 4.39i)13-s + 12.7·14-s + (6.15 + 6.15i)15-s + 18.3·16-s + 21.9i·17-s + ⋯
L(s)  = 1  + (0.768 + 0.768i)2-s + 0.605·3-s + 0.182i·4-s + (0.678 + 0.678i)5-s + (0.465 + 0.465i)6-s + (0.590 − 0.590i)7-s + (0.628 − 0.628i)8-s − 0.633·9-s + 1.04i·10-s + (−0.213 + 0.213i)11-s + 0.110i·12-s + (−0.941 + 0.337i)13-s + 0.908·14-s + (0.410 + 0.410i)15-s + 1.14·16-s + 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.670 - 0.741i$
Analytic conductor: \(3.89646\)
Root analytic conductor: \(1.97394\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1),\ 0.670 - 0.741i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.44582 + 1.08635i\)
\(L(\frac12)\) \(\approx\) \(2.44582 + 1.08635i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.34 - 2.34i)T \)
13 \( 1 + (12.2 - 4.39i)T \)
good2 \( 1 + (-1.53 - 1.53i)T + 4iT^{2} \)
3 \( 1 - 1.81T + 9T^{2} \)
5 \( 1 + (-3.39 - 3.39i)T + 25iT^{2} \)
7 \( 1 + (-4.13 + 4.13i)T - 49iT^{2} \)
17 \( 1 - 21.9iT - 289T^{2} \)
19 \( 1 + (9.45 + 9.45i)T + 361iT^{2} \)
23 \( 1 + 8.91iT - 529T^{2} \)
29 \( 1 - 12.0T + 841T^{2} \)
31 \( 1 + (9.11 + 9.11i)T + 961iT^{2} \)
37 \( 1 + (22.4 - 22.4i)T - 1.36e3iT^{2} \)
41 \( 1 + (32.0 + 32.0i)T + 1.68e3iT^{2} \)
43 \( 1 + 34.5iT - 1.84e3T^{2} \)
47 \( 1 + (-52.8 + 52.8i)T - 2.20e3iT^{2} \)
53 \( 1 - 75.1T + 2.80e3T^{2} \)
59 \( 1 + (46.6 - 46.6i)T - 3.48e3iT^{2} \)
61 \( 1 + 86.3T + 3.72e3T^{2} \)
67 \( 1 + (-52.1 - 52.1i)T + 4.48e3iT^{2} \)
71 \( 1 + (-81.1 - 81.1i)T + 5.04e3iT^{2} \)
73 \( 1 + (-46.9 + 46.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 13.1T + 6.24e3T^{2} \)
83 \( 1 + (13.2 + 13.2i)T + 6.88e3iT^{2} \)
89 \( 1 + (112. - 112. i)T - 7.92e3iT^{2} \)
97 \( 1 + (-105. - 105. i)T + 9.40e3iT^{2} \)
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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

−13.63780496668150529484540801866, −12.32069861655757669464545611119, −10.73947197209778593295004615929, −10.10143823636523333509527855201, −8.626854373185147991710889563386, −7.41393523580290353438125604334, −6.47977368256530492691095743524, −5.31500447879768559751132147533, −4.02392011847818150004222203633, −2.24300998719809526968810416361, 2.04378751220112190723125051238, 3.08455174551622599132846109776, 4.83151354466610804615154600238, 5.56327850780168951559932124633, 7.68104710413009695080465811089, 8.641083380728169971900179667006, 9.619784215433126394954850131221, 11.00876654004830311324377321947, 11.96462230450596893282349511122, 12.70285388864895621858653808425

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