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Properties

Label 2-143-143.98-c1-0-8
Degree $2$
Conductor $143$
Sign $-0.580 + 0.814i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.428 − 1.60i)2-s + (0.382 − 0.663i)3-s + (−0.647 + 0.373i)4-s + (−0.104 − 0.104i)5-s + (−1.22 − 0.328i)6-s + (0.542 − 2.02i)7-s + (−1.46 − 1.46i)8-s + (1.20 + 2.09i)9-s + (−0.122 + 0.211i)10-s + (−0.436 − 3.28i)11-s + 0.571i·12-s + (−2.52 + 2.57i)13-s − 3.47·14-s + (−0.109 + 0.0292i)15-s + (−2.46 + 4.27i)16-s + (1.88 + 3.25i)17-s + ⋯
L(s)  = 1  + (−0.303 − 1.13i)2-s + (0.221 − 0.382i)3-s + (−0.323 + 0.186i)4-s + (−0.0466 − 0.0466i)5-s + (−0.500 − 0.134i)6-s + (0.204 − 0.764i)7-s + (−0.519 − 0.519i)8-s + (0.402 + 0.696i)9-s + (−0.0386 + 0.0669i)10-s + (−0.131 − 0.991i)11-s + 0.165i·12-s + (−0.701 + 0.713i)13-s − 0.928·14-s + (−0.0281 + 0.00754i)15-s + (−0.617 + 1.06i)16-s + (0.456 + 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.580 + 0.814i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481005 - 0.933266i\)
\(L(\frac12)\) \(\approx\) \(0.481005 - 0.933266i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.436 + 3.28i)T \)
13 \( 1 + (2.52 - 2.57i)T \)
good2 \( 1 + (0.428 + 1.60i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-0.382 + 0.663i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.104 + 0.104i)T + 5iT^{2} \)
7 \( 1 + (-0.542 + 2.02i)T + (-6.06 - 3.5i)T^{2} \)
17 \( 1 + (-1.88 - 3.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.23 - 1.40i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.14 + 2.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.93 - 2.84i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.15 - 7.15i)T + 31iT^{2} \)
37 \( 1 + (-0.877 - 3.27i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.88 + 7.02i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.18 - 3.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.00268 + 0.00268i)T - 47iT^{2} \)
53 \( 1 + 5.02T + 53T^{2} \)
59 \( 1 + (4.54 + 1.21i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (11.8 - 6.83i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.04 - 2.15i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.26 - 4.72i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.92 + 6.92i)T + 73iT^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 + (1.45 - 1.45i)T - 83iT^{2} \)
89 \( 1 + (0.243 + 0.908i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.57 + 5.87i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.42713065749598081581189280668, −11.83240682204971395151187420918, −10.47371111099337960997040315167, −10.22095031057724035222455934122, −8.677027299918522935250487863219, −7.63094293767459125425573101433, −6.33542914228662080236702369748, −4.46036350866888791814671401462, −2.94301526422487487421839033258, −1.37049033948156422838342464171, 2.84888727818545376570018039246, 4.82685930928184984238077456982, 5.92865774620435051036718842745, 7.26673807235641110677896578522, 7.957002175876040977755654710346, 9.373012397397973033359386873022, 9.832525493195914732737584036928, 11.72294472035578757007630256952, 12.26235627754030428070922338933, 13.81740331076377351521694180154

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