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Properties

Label 2-143-13.12-c3-0-20
Degree $2$
Conductor $143$
Sign $-0.822 + 0.568i$
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  − 2.37i·2-s − 6.00·3-s + 2.37·4-s + 22.0i·5-s + 14.2i·6-s − 21.9i·7-s − 24.6i·8-s + 9.11·9-s + 52.1·10-s − 11i·11-s − 14.2·12-s + (−26.6 − 38.5i)13-s − 51.9·14-s − 132. i·15-s − 39.3·16-s + 45.0·17-s + ⋯
L(s)  = 1  − 0.838i·2-s − 1.15·3-s + 0.296·4-s + 1.96i·5-s + 0.969i·6-s − 1.18i·7-s − 1.08i·8-s + 0.337·9-s + 1.65·10-s − 0.301i·11-s − 0.343·12-s + (−0.568 − 0.822i)13-s − 0.992·14-s − 2.27i·15-s − 0.615·16-s + 0.642·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.822 + 0.568i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ -0.822 + 0.568i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.231280 - 0.741198i\)
\(L(\frac12)\) \(\approx\) \(0.231280 - 0.741198i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + 11iT \)
13 \( 1 + (26.6 + 38.5i)T \)
good2 \( 1 + 2.37iT - 8T^{2} \)
3 \( 1 + 6.00T + 27T^{2} \)
5 \( 1 - 22.0iT - 125T^{2} \)
7 \( 1 + 21.9iT - 343T^{2} \)
17 \( 1 - 45.0T + 4.91e3T^{2} \)
19 \( 1 + 118. iT - 6.85e3T^{2} \)
23 \( 1 + 37.3T + 1.21e4T^{2} \)
29 \( 1 + 137.T + 2.43e4T^{2} \)
31 \( 1 + 1.53iT - 2.97e4T^{2} \)
37 \( 1 + 236. iT - 5.06e4T^{2} \)
41 \( 1 + 72.6iT - 6.89e4T^{2} \)
43 \( 1 - 60.4T + 7.95e4T^{2} \)
47 \( 1 + 397. iT - 1.03e5T^{2} \)
53 \( 1 - 0.405T + 1.48e5T^{2} \)
59 \( 1 - 37.1iT - 2.05e5T^{2} \)
61 \( 1 - 308.T + 2.26e5T^{2} \)
67 \( 1 - 427. iT - 3.00e5T^{2} \)
71 \( 1 - 1.01e3iT - 3.57e5T^{2} \)
73 \( 1 - 186. iT - 3.89e5T^{2} \)
79 \( 1 + 831.T + 4.93e5T^{2} \)
83 \( 1 + 1.34e3iT - 5.71e5T^{2} \)
89 \( 1 - 840. iT - 7.04e5T^{2} \)
97 \( 1 - 1.23e3iT - 9.12e5T^{2} \)
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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

−11.70630622112552487228485944938, −11.14076046215547530298435933167, −10.52199951367210939332881110901, −9.993498338705094670074817471527, −7.37162410956662458146867990751, −6.91403035580225208106802403462, −5.75170443693887384370398635486, −3.75498923532814463808498562562, −2.63756297105332928600388624188, −0.42478710569235235336774899688, 1.71268647590554197032705856115, 4.71175327094393777378565141844, 5.53635664844134355430163605163, 6.11428251611826410076879461722, 7.78450243417164616877169788316, 8.700894393793755472621354121067, 9.753836292277369876550249665623, 11.50001773643219423254168953237, 12.12155279010270945243328212512, 12.56864471204279630473820842347

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