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Properties

Label 2-143-13.3-c3-0-20
Degree $2$
Conductor $143$
Sign $0.929 - 0.369i$
Analytic cond. $8.43727$
Root an. cond. $2.90469$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.56 + 2.70i)2-s + (0.268 + 0.465i)3-s + (−0.889 + 1.53i)4-s − 0.516·5-s + (−0.840 + 1.45i)6-s + (17.3 − 30.0i)7-s + 19.4·8-s + (13.3 − 23.1i)9-s + (−0.807 − 1.39i)10-s + (−5.5 − 9.52i)11-s − 0.955·12-s + (−18.8 + 42.9i)13-s + 108.·14-s + (−0.138 − 0.240i)15-s + (37.5 + 65.0i)16-s + (−57.0 + 98.8i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.957i)2-s + (0.0517 + 0.0896i)3-s + (−0.111 + 0.192i)4-s − 0.0461·5-s + (−0.0571 + 0.0990i)6-s + (0.935 − 1.61i)7-s + 0.859·8-s + (0.494 − 0.856i)9-s + (−0.0255 − 0.0442i)10-s + (−0.150 − 0.261i)11-s − 0.0229·12-s + (−0.401 + 0.915i)13-s + 2.06·14-s + (−0.00238 − 0.00413i)15-s + (0.586 + 1.01i)16-s + (−0.814 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.929 - 0.369i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 0.929 - 0.369i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.57077 + 0.492861i\)
\(L(\frac12)\) \(\approx\) \(2.57077 + 0.492861i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (5.5 + 9.52i)T \)
13 \( 1 + (18.8 - 42.9i)T \)
good2 \( 1 + (-1.56 - 2.70i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-0.268 - 0.465i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 0.516T + 125T^{2} \)
7 \( 1 + (-17.3 + 30.0i)T + (-171.5 - 297. i)T^{2} \)
17 \( 1 + (57.0 - 98.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-61.4 + 106. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-33.4 - 57.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (32.3 + 56.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 238.T + 2.97e4T^{2} \)
37 \( 1 + (14.1 + 24.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-154. - 266. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (123. - 213. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 165.T + 1.03e5T^{2} \)
53 \( 1 - 79.9T + 1.48e5T^{2} \)
59 \( 1 + (133. - 231. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (175. - 304. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (307. + 533. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (235. - 407. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 65.0T + 3.89e5T^{2} \)
79 \( 1 - 982.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + (-657. - 1.13e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (224. - 388. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.30157333902721194969179429019, −11.59287158926302088872460370377, −10.74026276777185979161967444330, −9.642457144610103777660949430809, −8.062753610895631434622890875428, −7.15320847498529912126774894283, −6.35997776002852977206187626038, −4.69413112369163487131907043475, −4.05399907890437417543221148518, −1.31203261468935639430208263668, 1.89586322612304206692715115392, 2.80292832006138755169756943800, 4.67117321816986232376377854708, 5.43545244929508786236921158926, 7.47079689090819306824407032669, 8.296833291881157485868397430763, 9.764319045708891392963030126844, 10.83319103517526911201327622254, 11.84942734140569528812684861004, 12.28907987417103638449074806279

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