↑

Properties

Label 2-153-17.15-c3-0-11
Degree 22
Conductor 153153
Sign −0.937−0.348i-0.937 - 0.348i
Analytic cond. 9.027299.02729
Root an. cond. 3.004543.00454
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.81 − 3.81i)2-s + 21.1i·4-s + (−1.33 + 0.554i)5-s + (8.84 + 3.66i)7-s + (50.0 − 50.0i)8-s + (7.22 + 2.99i)10-s + (−17.3 + 41.8i)11-s − 86.3i·13-s + (−19.7 − 47.7i)14-s − 212.·16-s + (−33.9 − 61.3i)17-s + (−25.5 − 25.5i)19-s + (−11.7 − 28.2i)20-s + (225. − 93.4i)22-s + (12.9 − 31.2i)23-s + ⋯
L(s)  = 1  + (−1.34 − 1.34i)2-s + 2.63i·4-s + (−0.119 + 0.0495i)5-s + (0.477 + 0.197i)7-s + (2.21 − 2.21i)8-s + (0.228 + 0.0945i)10-s + (−0.474 + 1.14i)11-s − 1.84i·13-s + (−0.377 − 0.910i)14-s − 3.32·16-s + (−0.484 − 0.875i)17-s + (−0.308 − 0.308i)19-s + (−0.130 − 0.315i)20-s + (2.18 − 0.905i)22-s + (0.117 − 0.283i)23-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=((−0.937−0.348i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+3/2) L(s)=((−0.937−0.348i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: −0.937−0.348i-0.937 - 0.348i
Analytic conductor: 9.027299.02729
Root analytic conductor: 3.004543.00454
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ153(100,⋅)\chi_{153} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 153, ( :3/2), −0.937−0.348i)(2,\ 153,\ (\ :3/2),\ -0.937 - 0.348i)

Particular Values

L(2)L(2) ≈\approx 0.0594237+0.330219i0.0594237 + 0.330219i
L(12)L(\frac12) ≈\approx 0.0594237+0.330219i0.0594237 + 0.330219i
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)
bad3 1 1
17 1+(33.9+61.3i)T 1 + (33.9 + 61.3i)T
good2 1+(3.81+3.81i)T+8iT2 1 + (3.81 + 3.81i)T + 8iT^{2}
5 1+(1.33−0.554i)T+(88.3−88.3i)T2 1 + (1.33 - 0.554i)T + (88.3 - 88.3i)T^{2}
7 1+(−8.84−3.66i)T+(242.+242.i)T2 1 + (-8.84 - 3.66i)T + (242. + 242. i)T^{2}
11 1+(17.3−41.8i)T+(−941.−941.i)T2 1 + (17.3 - 41.8i)T + (-941. - 941. i)T^{2}
13 1+86.3iT−2.19e3T2 1 + 86.3iT - 2.19e3T^{2}
19 1+(25.5+25.5i)T+6.85e3iT2 1 + (25.5 + 25.5i)T + 6.85e3iT^{2}
23 1+(−12.9+31.2i)T+(−8.60e3−8.60e3i)T2 1 + (-12.9 + 31.2i)T + (-8.60e3 - 8.60e3i)T^{2}
29 1+(−50.3+20.8i)T+(1.72e4−1.72e4i)T2 1 + (-50.3 + 20.8i)T + (1.72e4 - 1.72e4i)T^{2}
31 1+(−52.5−126.i)T+(−2.10e4+2.10e4i)T2 1 + (-52.5 - 126. i)T + (-2.10e4 + 2.10e4i)T^{2}
37 1+(166.+401.i)T+(−3.58e4+3.58e4i)T2 1 + (166. + 401. i)T + (-3.58e4 + 3.58e4i)T^{2}
41 1+(128.+53.3i)T+(4.87e4+4.87e4i)T2 1 + (128. + 53.3i)T + (4.87e4 + 4.87e4i)T^{2}
43 1+(231.−231.i)T−7.95e4iT2 1 + (231. - 231. i)T - 7.95e4iT^{2}
47 1+357.iT−1.03e5T2 1 + 357. iT - 1.03e5T^{2}
53 1+(118.+118.i)T+1.48e5iT2 1 + (118. + 118. i)T + 1.48e5iT^{2}
59 1+(216.−216.i)T−2.05e5iT2 1 + (216. - 216. i)T - 2.05e5iT^{2}
61 1+(198.+82.1i)T+(1.60e5+1.60e5i)T2 1 + (198. + 82.1i)T + (1.60e5 + 1.60e5i)T^{2}
67 1+287.T+3.00e5T2 1 + 287.T + 3.00e5T^{2}
71 1+(312.+754.i)T+(−2.53e5+2.53e5i)T2 1 + (312. + 754. i)T + (-2.53e5 + 2.53e5i)T^{2}
73 1+(−301.+124.i)T+(2.75e5−2.75e5i)T2 1 + (-301. + 124. i)T + (2.75e5 - 2.75e5i)T^{2}
79 1+(−230.+557.i)T+(−3.48e5−3.48e5i)T2 1 + (-230. + 557. i)T + (-3.48e5 - 3.48e5i)T^{2}
83 1+(601.+601.i)T+5.71e5iT2 1 + (601. + 601. i)T + 5.71e5iT^{2}
89 1−640.iT−7.04e5T2 1 - 640. iT - 7.04e5T^{2}
97 1+(−622.+257.i)T+(6.45e5−6.45e5i)T2 1 + (-622. + 257. i)T + (6.45e5 - 6.45e5i)T^{2}
show more
show less
   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

−11.74951109480005019722606259941, −10.71784335663947240372940868079, −10.10792036620660225545550465886, −9.029251613265213229227531301229, −8.024548329693957487176851250318, −7.24913798655740170135652059518, −4.90363133382532546523599388226, −3.20944298585490337317713464634, −2.00406833193304643026540936299, −0.25123913376458842603042520653, 1.59147520958125974176204236982, 4.54421166815279869654120632000, 5.99606026670883825490931636710, 6.78842688498728404020035709104, 8.093355855093347797220197283892, 8.594659803117958758813543440589, 9.710780639982677766075864883827, 10.75111218658183314723799818171, 11.61700451553911371007549712528, 13.63778067621914450210734873332

Graph of the ZZ-function along the critical line