↑

Properties

Label 2-405-135.122-c1-0-13
Degree 22
Conductor 405405
Sign −0.251+0.967i-0.251 + 0.967i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.377 + 0.264i)2-s + (−0.611 − 1.67i)4-s + (−0.538 + 2.17i)5-s + (−1.52 − 3.26i)7-s + (0.451 − 1.68i)8-s + (−0.777 + 0.676i)10-s + (−2.85 − 3.40i)11-s + (0.226 + 0.323i)13-s + (0.288 − 1.63i)14-s + (−2.12 + 1.78i)16-s + (−1.03 − 3.86i)17-s + (1.05 − 0.610i)19-s + (3.97 − 0.421i)20-s + (−0.178 − 2.03i)22-s + (3.57 + 1.66i)23-s + ⋯
L(s)  = 1  + (0.266 + 0.186i)2-s + (−0.305 − 0.839i)4-s + (−0.241 + 0.970i)5-s + (−0.575 − 1.23i)7-s + (0.159 − 0.596i)8-s + (−0.245 + 0.214i)10-s + (−0.860 − 1.02i)11-s + (0.0627 + 0.0896i)13-s + (0.0770 − 0.436i)14-s + (−0.530 + 0.445i)16-s + (−0.251 − 0.936i)17-s + (0.242 − 0.140i)19-s + (0.888 − 0.0942i)20-s + (−0.0380 − 0.434i)22-s + (0.746 + 0.347i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.251+0.967i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.251+0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.251+0.967i-0.251 + 0.967i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(152,⋅)\chi_{405} (152, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), −0.251+0.967i)(2,\ 405,\ (\ :1/2),\ -0.251 + 0.967i)

Particular Values

L(1)L(1) ≈\approx 0.585975−0.757425i0.585975 - 0.757425i
L(12)L(\frac12) ≈\approx 0.585975−0.757425i0.585975 - 0.757425i
L(32)L(\frac{3}{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
5 1+(0.538−2.17i)T 1 + (0.538 - 2.17i)T
good2 1+(−0.377−0.264i)T+(0.684+1.87i)T2 1 + (-0.377 - 0.264i)T + (0.684 + 1.87i)T^{2}
7 1+(1.52+3.26i)T+(−4.49+5.36i)T2 1 + (1.52 + 3.26i)T + (-4.49 + 5.36i)T^{2}
11 1+(2.85+3.40i)T+(−1.91+10.8i)T2 1 + (2.85 + 3.40i)T + (-1.91 + 10.8i)T^{2}
13 1+(−0.226−0.323i)T+(−4.44+12.2i)T2 1 + (-0.226 - 0.323i)T + (-4.44 + 12.2i)T^{2}
17 1+(1.03+3.86i)T+(−14.7+8.5i)T2 1 + (1.03 + 3.86i)T + (-14.7 + 8.5i)T^{2}
19 1+(−1.05+0.610i)T+(9.5−16.4i)T2 1 + (-1.05 + 0.610i)T + (9.5 - 16.4i)T^{2}
23 1+(−3.57−1.66i)T+(14.7+17.6i)T2 1 + (-3.57 - 1.66i)T + (14.7 + 17.6i)T^{2}
29 1+(0.885+5.02i)T+(−27.2+9.91i)T2 1 + (0.885 + 5.02i)T + (-27.2 + 9.91i)T^{2}
31 1+(−9.19+3.34i)T+(23.7−19.9i)T2 1 + (-9.19 + 3.34i)T + (23.7 - 19.9i)T^{2}
37 1+(10.6−2.84i)T+(32.0−18.5i)T2 1 + (10.6 - 2.84i)T + (32.0 - 18.5i)T^{2}
41 1+(−1.80−0.318i)T+(38.5+14.0i)T2 1 + (-1.80 - 0.318i)T + (38.5 + 14.0i)T^{2}
43 1+(0.218−2.49i)T+(−42.3−7.46i)T2 1 + (0.218 - 2.49i)T + (-42.3 - 7.46i)T^{2}
47 1+(1.18−0.554i)T+(30.2−36.0i)T2 1 + (1.18 - 0.554i)T + (30.2 - 36.0i)T^{2}
53 1+(−3.53−3.53i)T+53iT2 1 + (-3.53 - 3.53i)T + 53iT^{2}
59 1+(−0.467−0.391i)T+(10.2+58.1i)T2 1 + (-0.467 - 0.391i)T + (10.2 + 58.1i)T^{2}
61 1+(1.87+0.683i)T+(46.7+39.2i)T2 1 + (1.87 + 0.683i)T + (46.7 + 39.2i)T^{2}
67 1+(−3.34+2.34i)T+(22.9−62.9i)T2 1 + (-3.34 + 2.34i)T + (22.9 - 62.9i)T^{2}
71 1+(−10.9−6.30i)T+(35.5+61.4i)T2 1 + (-10.9 - 6.30i)T + (35.5 + 61.4i)T^{2}
73 1+(−4.00−1.07i)T+(63.2+36.5i)T2 1 + (-4.00 - 1.07i)T + (63.2 + 36.5i)T^{2}
79 1+(−11.8+2.09i)T+(74.2−27.0i)T2 1 + (-11.8 + 2.09i)T + (74.2 - 27.0i)T^{2}
83 1+(1.54−2.20i)T+(−28.3−77.9i)T2 1 + (1.54 - 2.20i)T + (-28.3 - 77.9i)T^{2}
89 1+(1.23+2.13i)T+(−44.5+77.0i)T2 1 + (1.23 + 2.13i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.567−0.0496i)T+(95.5+16.8i)T2 1 + (-0.567 - 0.0496i)T + (95.5 + 16.8i)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

−10.82175044704897995988726746509, −10.22887972101213060854174270783, −9.450067899713204705018957459166, −8.026582565352648603309367531581, −7.01070941395287315343594026826, −6.36059700884820081047515103642, −5.19776502441649497411147221984, −3.98460304482037544208547070630, −2.88649881437348576015132234973, −0.56512280886366448659108174074, 2.21943262317393436463784996337, 3.45895406465383226832284665587, 4.74858765120266206437441683430, 5.41626994094993164834217000152, 6.89526927234373306420567084385, 8.118376735343112830803949560837, 8.681733218238475487067904060453, 9.490105915107510840661911872545, 10.67033964336532218973771095529, 12.01550434018311871671591044748

Graph of the ZZ-function along the critical line