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Properties

Label 2-160-160.107-c3-0-24
Degree $2$
Conductor $160$
Sign $0.540 + 0.841i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
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  + (−2.77 + 0.564i)2-s + (−8.63 − 3.57i)3-s + (7.36 − 3.12i)4-s + (10.3 − 4.22i)5-s + (25.9 + 5.03i)6-s − 6.56·7-s + (−18.6 + 12.8i)8-s + (42.6 + 42.6i)9-s + (−26.2 + 17.5i)10-s + (−14.1 + 34.1i)11-s + (−74.7 + 0.685i)12-s + (1.64 + 0.679i)13-s + (18.1 − 3.70i)14-s + (−104. − 0.516i)15-s + (44.4 − 46.0i)16-s + (57.8 − 57.8i)17-s + ⋯
L(s)  = 1  + (−0.979 + 0.199i)2-s + (−1.66 − 0.688i)3-s + (0.920 − 0.391i)4-s + (0.925 − 0.378i)5-s + (1.76 + 0.342i)6-s − 0.354·7-s + (−0.823 + 0.566i)8-s + (1.57 + 1.57i)9-s + (−0.831 + 0.555i)10-s + (−0.387 + 0.936i)11-s + (−1.79 + 0.0164i)12-s + (0.0350 + 0.0145i)13-s + (0.347 − 0.0707i)14-s + (−1.79 − 0.00889i)15-s + (0.694 − 0.719i)16-s + (0.826 − 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.540 + 0.841i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ 0.540 + 0.841i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.558963 - 0.305093i\)
\(L(\frac12)\) \(\approx\) \(0.558963 - 0.305093i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.77 - 0.564i)T \)
5 \( 1 + (-10.3 + 4.22i)T \)
good3 \( 1 + (8.63 + 3.57i)T + (19.0 + 19.0i)T^{2} \)
7 \( 1 + 6.56T + 343T^{2} \)
11 \( 1 + (14.1 - 34.1i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-1.64 - 0.679i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + (-57.8 + 57.8i)T - 4.91e3iT^{2} \)
19 \( 1 + (-38.5 - 93.0i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 - 50.4T + 1.21e4T^{2} \)
29 \( 1 + (6.86 + 16.5i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 320. iT - 2.97e4T^{2} \)
37 \( 1 + (170. - 70.7i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-225. + 225. i)T - 6.89e4iT^{2} \)
43 \( 1 + (89.0 + 214. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + (-286. - 286. i)T + 1.03e5iT^{2} \)
53 \( 1 + (5.39 - 2.23i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-316. + 763. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-223. + 92.4i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-113. + 273. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-492. + 492. i)T - 3.57e5iT^{2} \)
73 \( 1 - 734. iT - 3.89e5T^{2} \)
79 \( 1 - 590. iT - 4.93e5T^{2} \)
83 \( 1 + (185. - 448. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-693. + 693. i)T - 7.04e5iT^{2} \)
97 \( 1 + (-66.1 - 66.1i)T + 9.12e5iT^{2} \)
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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

−12.20422541106093804585751102706, −11.19308714740896064326364781295, −10.11098756423439073233963834029, −9.594685718953405525920427782357, −7.80799582539996816903340242265, −6.91024756019532027786115452907, −5.89345481887195310491114315359, −5.19507762547514754446770837105, −1.99094204407716303886749857792, −0.67930803576026470272516382888, 0.987467825959502615477270979458, 3.22541997383520258072598414398, 5.29298330889769547088171865566, 6.15111448315136363920828042264, 7.03643170011752793797507281712, 8.852469559406924870428689767768, 9.892829703849363079406917380146, 10.57670440856979777124357713316, 11.14678639166160323253153089913, 12.20129513491903541557861759903

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