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Properties

Label 2-143-13.12-c3-0-7
Degree $2$
Conductor $143$
Sign $0.265 + 0.964i$
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  + 4.98i·2-s − 4.47·3-s − 16.8·4-s + 20.1i·5-s − 22.2i·6-s + 23.8i·7-s − 44.2i·8-s − 7.01·9-s − 100.·10-s + 11i·11-s + 75.4·12-s + (45.1 − 12.4i)13-s − 118.·14-s − 90.1i·15-s + 85.7·16-s + 111.·17-s + ⋯
L(s)  = 1  + 1.76i·2-s − 0.860·3-s − 2.10·4-s + 1.80i·5-s − 1.51i·6-s + 1.28i·7-s − 1.95i·8-s − 0.259·9-s − 3.17·10-s + 0.301i·11-s + 1.81·12-s + (0.964 − 0.265i)13-s − 2.27·14-s − 1.55i·15-s + 1.34·16-s + 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.265 + 0.964i$
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: \((2,\ 143,\ (\ :3/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.700526 - 0.533732i\)
\(L(\frac12)\) \(\approx\) \(0.700526 - 0.533732i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-45.1 + 12.4i)T \)
good2 \( 1 - 4.98iT - 8T^{2} \)
3 \( 1 + 4.47T + 27T^{2} \)
5 \( 1 - 20.1iT - 125T^{2} \)
7 \( 1 - 23.8iT - 343T^{2} \)
17 \( 1 - 111.T + 4.91e3T^{2} \)
19 \( 1 - 3.85iT - 6.85e3T^{2} \)
23 \( 1 - 76.9T + 1.21e4T^{2} \)
29 \( 1 + 85.3T + 2.43e4T^{2} \)
31 \( 1 - 244. iT - 2.97e4T^{2} \)
37 \( 1 - 180. iT - 5.06e4T^{2} \)
41 \( 1 + 470. iT - 6.89e4T^{2} \)
43 \( 1 - 194.T + 7.95e4T^{2} \)
47 \( 1 - 287. iT - 1.03e5T^{2} \)
53 \( 1 + 564.T + 1.48e5T^{2} \)
59 \( 1 + 50.4iT - 2.05e5T^{2} \)
61 \( 1 - 334.T + 2.26e5T^{2} \)
67 \( 1 + 199. iT - 3.00e5T^{2} \)
71 \( 1 - 152. iT - 3.57e5T^{2} \)
73 \( 1 + 245. iT - 3.89e5T^{2} \)
79 \( 1 + 281.T + 4.93e5T^{2} \)
83 \( 1 - 434. iT - 5.71e5T^{2} \)
89 \( 1 + 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 - 802. iT - 9.12e5T^{2} \)
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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

−14.08109677314745317578310995757, −12.49882965297087309809258282674, −11.37448591783937492209326239429, −10.35072981882014471447330262217, −8.995230098288133383238116449669, −7.82064211152621996431712988529, −6.75759375290354300212467241963, −5.97117136881842936008742715958, −5.35528223381478346393553748150, −3.20548065928233269590473574761, 0.59040278264005506085686475514, 1.22685908910461791453681570525, 3.65987301801382978291271502287, 4.65543568913645690562473365837, 5.72768297567755922839544234680, 8.012158849398194321440727535262, 9.103232024432366155371878966867, 10.03108197416015181847691417857, 11.12172912034569618465137781537, 11.67704453706696592545011045041

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