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Properties

Label 2-160-160.107-c3-0-38
Degree $2$
Conductor $160$
Sign $-0.289 - 0.957i$
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  + (2.06 + 1.92i)2-s + (6.09 + 2.52i)3-s + (0.553 + 7.98i)4-s + (1.73 + 11.0i)5-s + (7.73 + 16.9i)6-s + 13.5·7-s + (−14.2 + 17.5i)8-s + (11.6 + 11.6i)9-s + (−17.7 + 26.1i)10-s + (26.3 − 63.6i)11-s + (−16.7 + 50.0i)12-s + (−62.3 − 25.8i)13-s + (27.9 + 26.1i)14-s + (−17.2 + 71.6i)15-s + (−63.3 + 8.82i)16-s + (8.04 − 8.04i)17-s + ⋯
L(s)  = 1  + (0.731 + 0.682i)2-s + (1.17 + 0.485i)3-s + (0.0691 + 0.997i)4-s + (0.155 + 0.987i)5-s + (0.526 + 1.15i)6-s + 0.730·7-s + (−0.630 + 0.776i)8-s + (0.432 + 0.432i)9-s + (−0.560 + 0.828i)10-s + (0.723 − 1.74i)11-s + (−0.403 + 1.20i)12-s + (−1.32 − 0.550i)13-s + (0.534 + 0.498i)14-s + (−0.297 + 1.23i)15-s + (−0.990 + 0.137i)16-s + (0.114 − 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.289 - 0.957i$
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: \((2,\ 160,\ (\ :3/2),\ -0.289 - 0.957i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.19477 + 2.95653i\)
\(L(\frac12)\) \(\approx\) \(2.19477 + 2.95653i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.06 - 1.92i)T \)
5 \( 1 + (-1.73 - 11.0i)T \)
good3 \( 1 + (-6.09 - 2.52i)T + (19.0 + 19.0i)T^{2} \)
7 \( 1 - 13.5T + 343T^{2} \)
11 \( 1 + (-26.3 + 63.6i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (62.3 + 25.8i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + (-8.04 + 8.04i)T - 4.91e3iT^{2} \)
19 \( 1 + (6.34 + 15.3i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 - 125.T + 1.21e4T^{2} \)
29 \( 1 + (-84.2 - 203. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 3.81iT - 2.97e4T^{2} \)
37 \( 1 + (-240. + 99.7i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (111. - 111. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-88.5 - 213. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + (257. + 257. i)T + 1.03e5iT^{2} \)
53 \( 1 + (321. - 133. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (111. - 270. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-376. + 155. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (104. - 251. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-229. + 229. i)T - 3.57e5iT^{2} \)
73 \( 1 + 1.02e3iT - 3.89e5T^{2} \)
79 \( 1 - 851. iT - 4.93e5T^{2} \)
83 \( 1 + (-519. + 1.25e3i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-54.2 + 54.2i)T - 7.04e5iT^{2} \)
97 \( 1 + (185. + 185. i)T + 9.12e5iT^{2} \)
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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

−13.18034500587462579672752814334, −11.71832881015122313030040586209, −10.84955462196240801991892967172, −9.372012503457869692957520138230, −8.420381821513221357869507285230, −7.53937237806882340116624452988, −6.32269963120431774239483241784, −4.95192673040729036793887220545, −3.44305657157538611965750657102, −2.77277491979899267289218390022, 1.53747094650937594885297495717, 2.38379561023602020587499920548, 4.27320694465833957470762945503, 5.01091230520626870662250652435, 6.88411752510815067309413820180, 8.016515037913857894360718663073, 9.312515037489238996352436452962, 9.800642515360713195579099823523, 11.54770858585236920855311981912, 12.37702327073262190958148455540

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