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Properties

Label 2-160-160.43-c3-0-13
Degree $2$
Conductor $160$
Sign $0.0970 - 0.995i$
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  + (−1.96 + 2.03i)2-s + (3.68 − 8.88i)3-s + (−0.316 − 7.99i)4-s + (−6.53 + 9.07i)5-s + (10.9 + 24.9i)6-s + 30.1i·7-s + (16.9 + 15.0i)8-s + (−46.3 − 46.3i)9-s + (−5.69 − 31.1i)10-s + (−7.33 + 17.7i)11-s + (−72.1 − 26.6i)12-s + (−3.98 + 9.61i)13-s + (−61.4 − 59.0i)14-s + (56.5 + 91.4i)15-s + (−63.7 + 5.05i)16-s + (60.6 + 60.6i)17-s + ⋯
L(s)  = 1  + (−0.692 + 0.720i)2-s + (0.708 − 1.71i)3-s + (−0.0395 − 0.999i)4-s + (−0.584 + 0.811i)5-s + (0.742 + 1.69i)6-s + 1.62i·7-s + (0.747 + 0.663i)8-s + (−1.71 − 1.71i)9-s + (−0.179 − 0.983i)10-s + (−0.201 + 0.485i)11-s + (−1.73 − 0.640i)12-s + (−0.0850 + 0.205i)13-s + (−1.17 − 1.12i)14-s + (0.973 + 1.57i)15-s + (−0.996 + 0.0790i)16-s + (0.864 + 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0970 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0970 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.0970 - 0.995i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ 0.0970 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.725340 + 0.658048i\)
\(L(\frac12)\) \(\approx\) \(0.725340 + 0.658048i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 - 2.03i)T \)
5 \( 1 + (6.53 - 9.07i)T \)
good3 \( 1 + (-3.68 + 8.88i)T + (-19.0 - 19.0i)T^{2} \)
7 \( 1 - 30.1iT - 343T^{2} \)
11 \( 1 + (7.33 - 17.7i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (3.98 - 9.61i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + (-60.6 - 60.6i)T + 4.91e3iT^{2} \)
19 \( 1 + (-20.4 - 49.2i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + 56.6iT - 1.21e4T^{2} \)
29 \( 1 + (-107. - 260. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 158. iT - 2.97e4T^{2} \)
37 \( 1 + (65.1 + 157. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-104. + 104. i)T - 6.89e4iT^{2} \)
43 \( 1 + (243. - 100. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + (-176. + 176. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-132. - 319. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (247. - 597. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (402. - 166. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (330. + 136. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-132. + 132. i)T - 3.57e5iT^{2} \)
73 \( 1 + 13.2T + 3.89e5T^{2} \)
79 \( 1 + 325. iT - 4.93e5T^{2} \)
83 \( 1 + (-585. - 242. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-61.9 + 61.9i)T - 7.04e5iT^{2} \)
97 \( 1 + (-123. + 123. i)T - 9.12e5iT^{2} \)
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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

−12.38125338202674974441357767542, −12.09838823715606462057238586955, −10.55566605707762409377132262239, −9.021142978871436921740664712486, −8.332420502967429489646247649420, −7.48660904557178406818153746249, −6.61696373624752850701908043216, −5.67430283787259288089523925136, −2.88371529065317735877745743119, −1.67010961959143753273102364046, 0.54341939906743425996352568823, 3.13682088966351745254670094628, 4.02621744357368157602981806570, 4.87107588511979509277480586535, 7.62892305363259947621615414480, 8.245543133438560393640534052087, 9.473894132431111852480497611926, 9.966335151486651076309860829922, 10.95256439159242383919932382578, 11.70198276577217733573266416352

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