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Properties

Label 2-160-160.67-c3-0-43
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: $\chi_{160} (67, \cdot )$
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

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

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