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Properties

Label 2-160-160.67-c3-0-43
Degree 22
Conductor 160160
Sign 0.0970+0.995i0.0970 + 0.995i
Analytic cond. 9.440309.44030
Root an. cond. 3.072503.07250
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(160s/2 ΓC(s) L(s)=((0.0970+0.995i) Λ‾(4−s)\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}
Λ(s)=(160s/2 ΓC(s+3/2) L(s)=((0.0970+0.995i) Λ‾(1−s)\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: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: 0.0970+0.995i0.0970 + 0.995i
Analytic conductor: 9.440309.44030
Root analytic conductor: 3.072503.07250
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ160(67,⋅)\chi_{160} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 160, ( :3/2), 0.0970+0.995i)(2,\ 160,\ (\ :3/2),\ 0.0970 + 0.995i)

Particular Values

L(2)L(2) ≈\approx 0.725340−0.658048i0.725340 - 0.658048i
L(12)L(\frac12) ≈\approx 0.725340−0.658048i0.725340 - 0.658048i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.96+2.03i)T 1 + (1.96 + 2.03i)T
5 1+(6.53+9.07i)T 1 + (6.53 + 9.07i)T
good3 1+(−3.68−8.88i)T+(−19.0+19.0i)T2 1 + (-3.68 - 8.88i)T + (-19.0 + 19.0i)T^{2}
7 1+30.1iT−343T2 1 + 30.1iT - 343T^{2}
11 1+(7.33+17.7i)T+(−941.+941.i)T2 1 + (7.33 + 17.7i)T + (-941. + 941. i)T^{2}
13 1+(3.98+9.61i)T+(−1.55e3+1.55e3i)T2 1 + (3.98 + 9.61i)T + (-1.55e3 + 1.55e3i)T^{2}
17 1+(−60.6+60.6i)T−4.91e3iT2 1 + (-60.6 + 60.6i)T - 4.91e3iT^{2}
19 1+(−20.4+49.2i)T+(−4.85e3−4.85e3i)T2 1 + (-20.4 + 49.2i)T + (-4.85e3 - 4.85e3i)T^{2}
23 1−56.6iT−1.21e4T2 1 - 56.6iT - 1.21e4T^{2}
29 1+(−107.+260.i)T+(−1.72e4−1.72e4i)T2 1 + (-107. + 260. i)T + (-1.72e4 - 1.72e4i)T^{2}
31 1+158.iT−2.97e4T2 1 + 158. iT - 2.97e4T^{2}
37 1+(65.1−157.i)T+(−3.58e4−3.58e4i)T2 1 + (65.1 - 157. i)T + (-3.58e4 - 3.58e4i)T^{2}
41 1+(−104.−104.i)T+6.89e4iT2 1 + (-104. - 104. i)T + 6.89e4iT^{2}
43 1+(243.+100.i)T+(5.62e4+5.62e4i)T2 1 + (243. + 100. i)T + (5.62e4 + 5.62e4i)T^{2}
47 1+(−176.−176.i)T+1.03e5iT2 1 + (-176. - 176. i)T + 1.03e5iT^{2}
53 1+(−132.+319.i)T+(−1.05e5−1.05e5i)T2 1 + (-132. + 319. i)T + (-1.05e5 - 1.05e5i)T^{2}
59 1+(247.+597.i)T+(−1.45e5+1.45e5i)T2 1 + (247. + 597. i)T + (-1.45e5 + 1.45e5i)T^{2}
61 1+(402.+166.i)T+(1.60e5+1.60e5i)T2 1 + (402. + 166. i)T + (1.60e5 + 1.60e5i)T^{2}
67 1+(330.−136.i)T+(2.12e5−2.12e5i)T2 1 + (330. - 136. i)T + (2.12e5 - 2.12e5i)T^{2}
71 1+(−132.−132.i)T+3.57e5iT2 1 + (-132. - 132. i)T + 3.57e5iT^{2}
73 1+13.2T+3.89e5T2 1 + 13.2T + 3.89e5T^{2}
79 1−325.iT−4.93e5T2 1 - 325. iT - 4.93e5T^{2}
83 1+(−585.+242.i)T+(4.04e5−4.04e5i)T2 1 + (-585. + 242. i)T + (4.04e5 - 4.04e5i)T^{2}
89 1+(−61.9−61.9i)T+7.04e5iT2 1 + (-61.9 - 61.9i)T + 7.04e5iT^{2}
97 1+(−123.−123.i)T+9.12e5iT2 1 + (-123. - 123. i)T + 9.12e5iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line