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Properties

Label 2-143-13.9-c3-0-24
Degree $2$
Conductor $143$
Sign $0.350 + 0.936i$
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  + (1.58 − 2.74i)2-s + (0.300 − 0.520i)3-s + (−1.04 − 1.80i)4-s + 12.4·5-s + (−0.953 − 1.65i)6-s + (−5.31 − 9.20i)7-s + 18.7·8-s + (13.3 + 23.0i)9-s + (19.7 − 34.2i)10-s + (−5.5 + 9.52i)11-s − 1.24·12-s + (19.1 − 42.7i)13-s − 33.7·14-s + (3.74 − 6.48i)15-s + (38.1 − 66.0i)16-s + (31.4 + 54.5i)17-s + ⋯
L(s)  = 1  + (0.561 − 0.972i)2-s + (0.0577 − 0.100i)3-s + (−0.130 − 0.225i)4-s + 1.11·5-s + (−0.0648 − 0.112i)6-s + (−0.286 − 0.496i)7-s + 0.830·8-s + (0.493 + 0.854i)9-s + (0.625 − 1.08i)10-s + (−0.150 + 0.261i)11-s − 0.0300·12-s + (0.408 − 0.912i)13-s − 0.644·14-s + (0.0644 − 0.111i)15-s + (0.596 − 1.03i)16-s + (0.449 + 0.778i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.350 + 0.936i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 0.350 + 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.35274 - 1.63241i\)
\(L(\frac12)\) \(\approx\) \(2.35274 - 1.63241i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-19.1 + 42.7i)T \)
good2 \( 1 + (-1.58 + 2.74i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-0.300 + 0.520i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 12.4T + 125T^{2} \)
7 \( 1 + (5.31 + 9.20i)T + (-171.5 + 297. i)T^{2} \)
17 \( 1 + (-31.4 - 54.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (70.2 + 121. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (34.6 - 60.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-69.1 + 119. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 168.T + 2.97e4T^{2} \)
37 \( 1 + (118. - 204. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (103. - 179. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-150. - 261. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 343.T + 1.03e5T^{2} \)
53 \( 1 - 306.T + 1.48e5T^{2} \)
59 \( 1 + (-133. - 230. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-366. - 635. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-90.3 + 156. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-109. - 189. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 817.T + 3.89e5T^{2} \)
79 \( 1 + 289.T + 4.93e5T^{2} \)
83 \( 1 + 971.T + 5.71e5T^{2} \)
89 \( 1 + (702. - 1.21e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (860. + 1.48e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.87579366009895313574175586519, −11.37748982959055636323317797571, −10.39060861172067753014297277442, −9.952439044931908693323448344502, −8.231946447680275046741009788651, −7.00965661341162621193559151796, −5.53679600725714776781331855089, −4.27090530391306454306363928344, −2.75602503189481421836736881328, −1.55593679958473792635623860782, 1.80769016957894011118965234571, 3.90162705242188596682943894614, 5.44424445853966729719539921677, 6.19070471410684415062346342687, 7.04931266932761746517884649671, 8.649389018143628441628275737391, 9.685411920332615348845561954114, 10.59198021381906236329458037197, 12.18013541971104895383676601744, 13.06616776534274631283392122682

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