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Properties

Label 2-160-160.107-c3-0-32
Degree $2$
Conductor $160$
Sign $0.991 + 0.127i$
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  + (2.82 − 0.0971i)2-s + (−7.69 − 3.18i)3-s + (7.98 − 0.549i)4-s + (5.57 + 9.69i)5-s + (−22.0 − 8.26i)6-s − 1.12·7-s + (22.5 − 2.32i)8-s + (29.9 + 29.9i)9-s + (16.6 + 26.8i)10-s + (3.90 − 9.42i)11-s + (−63.1 − 21.2i)12-s + (51.7 + 21.4i)13-s + (−3.18 + 0.109i)14-s + (−11.9 − 92.3i)15-s + (63.3 − 8.76i)16-s + (86.0 − 86.0i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0343i)2-s + (−1.48 − 0.613i)3-s + (0.997 − 0.0686i)4-s + (0.498 + 0.866i)5-s + (−1.50 − 0.562i)6-s − 0.0607·7-s + (0.994 − 0.102i)8-s + (1.10 + 1.10i)9-s + (0.527 + 0.849i)10-s + (0.107 − 0.258i)11-s + (−1.51 − 0.510i)12-s + (1.10 + 0.457i)13-s + (−0.0607 + 0.00208i)14-s + (−0.206 − 1.58i)15-s + (0.990 − 0.136i)16-s + (1.22 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.991 + 0.127i$
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: \((2,\ 160,\ (\ :3/2),\ 0.991 + 0.127i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.31989 - 0.148062i\)
\(L(\frac12)\) \(\approx\) \(2.31989 - 0.148062i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 + 0.0971i)T \)
5 \( 1 + (-5.57 - 9.69i)T \)
good3 \( 1 + (7.69 + 3.18i)T + (19.0 + 19.0i)T^{2} \)
7 \( 1 + 1.12T + 343T^{2} \)
11 \( 1 + (-3.90 + 9.42i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-51.7 - 21.4i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + (-86.0 + 86.0i)T - 4.91e3iT^{2} \)
19 \( 1 + (-3.08 - 7.43i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 - 39.4T + 1.21e4T^{2} \)
29 \( 1 + (-32.0 - 77.3i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 56.9iT - 2.97e4T^{2} \)
37 \( 1 + (-219. + 91.0i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (299. - 299. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-90.2 - 217. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + (284. + 284. i)T + 1.03e5iT^{2} \)
53 \( 1 + (39.8 - 16.5i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (255. - 615. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (515. - 213. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-273. + 659. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-738. + 738. i)T - 3.57e5iT^{2} \)
73 \( 1 + 674. iT - 3.89e5T^{2} \)
79 \( 1 - 38.9iT - 4.93e5T^{2} \)
83 \( 1 + (543. - 1.31e3i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-81.7 + 81.7i)T - 7.04e5iT^{2} \)
97 \( 1 + (489. + 489. i)T + 9.12e5iT^{2} \)
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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

−12.32408513082062269966361542511, −11.40718459475792579602465819669, −10.99856252838674395825866907626, −9.831609721552611984047732108227, −7.59374521507428059745065340404, −6.58350973239325885944938628046, −6.01272121195551157711034090870, −4.98816939128067867189135805903, −3.20084764203953319236116531331, −1.35543302891567690395489620175, 1.26026379767860053488982056380, 3.75956802949831181340112829736, 4.91692727159437758918793026925, 5.73206067830902893250121856801, 6.42213739373983256787300838087, 8.180448407279085355066560405458, 9.872031171514459123747092998920, 10.64775508351047163992064934189, 11.59540662973474541664382121597, 12.49478592787251161077586305125

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