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Properties

Label 2-405-135.83-c1-0-11
Degree 22
Conductor 405405
Sign −0.318+0.948i-0.318 + 0.948i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.94 + 1.36i)2-s + (1.25 − 3.43i)4-s + (−0.278 − 2.21i)5-s + (−1.02 + 2.19i)7-s + (1.02 + 3.81i)8-s + (3.57 + 3.94i)10-s + (−0.00275 + 0.00327i)11-s + (0.792 − 1.13i)13-s + (−1.00 − 5.68i)14-s + (−1.59 − 1.33i)16-s + (−0.510 + 1.90i)17-s + (−6.69 − 3.86i)19-s + (−7.98 − 1.81i)20-s + (0.000887 − 0.0101i)22-s + (−6.70 + 3.12i)23-s + ⋯
L(s)  = 1  + (−1.37 + 0.964i)2-s + (0.626 − 1.71i)4-s + (−0.124 − 0.992i)5-s + (−0.387 + 0.830i)7-s + (0.361 + 1.34i)8-s + (1.12 + 1.24i)10-s + (−0.000829 + 0.000988i)11-s + (0.219 − 0.313i)13-s + (−0.267 − 1.51i)14-s + (−0.398 − 0.334i)16-s + (−0.123 + 0.461i)17-s + (−1.53 − 0.886i)19-s + (−1.78 − 0.406i)20-s + (0.000189 − 0.00216i)22-s + (−1.39 + 0.651i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.318+0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.318+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.318+0.948i-0.318 + 0.948i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(8,⋅)\chi_{405} (8, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), −0.318+0.948i)(2,\ 405,\ (\ :1/2),\ -0.318 + 0.948i)

Particular Values

L(1)L(1) ≈\approx 0.0966400−0.134368i0.0966400 - 0.134368i
L(12)L(\frac12) ≈\approx 0.0966400−0.134368i0.0966400 - 0.134368i
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+(0.278+2.21i)T 1 + (0.278 + 2.21i)T
good2 1+(1.94−1.36i)T+(0.684−1.87i)T2 1 + (1.94 - 1.36i)T + (0.684 - 1.87i)T^{2}
7 1+(1.02−2.19i)T+(−4.49−5.36i)T2 1 + (1.02 - 2.19i)T + (-4.49 - 5.36i)T^{2}
11 1+(0.00275−0.00327i)T+(−1.91−10.8i)T2 1 + (0.00275 - 0.00327i)T + (-1.91 - 10.8i)T^{2}
13 1+(−0.792+1.13i)T+(−4.44−12.2i)T2 1 + (-0.792 + 1.13i)T + (-4.44 - 12.2i)T^{2}
17 1+(0.510−1.90i)T+(−14.7−8.5i)T2 1 + (0.510 - 1.90i)T + (-14.7 - 8.5i)T^{2}
19 1+(6.69+3.86i)T+(9.5+16.4i)T2 1 + (6.69 + 3.86i)T + (9.5 + 16.4i)T^{2}
23 1+(6.70−3.12i)T+(14.7−17.6i)T2 1 + (6.70 - 3.12i)T + (14.7 - 17.6i)T^{2}
29 1+(−1.74+9.87i)T+(−27.2−9.91i)T2 1 + (-1.74 + 9.87i)T + (-27.2 - 9.91i)T^{2}
31 1+(5.62+2.04i)T+(23.7+19.9i)T2 1 + (5.62 + 2.04i)T + (23.7 + 19.9i)T^{2}
37 1+(−1.68−0.451i)T+(32.0+18.5i)T2 1 + (-1.68 - 0.451i)T + (32.0 + 18.5i)T^{2}
41 1+(−2.95+0.520i)T+(38.5−14.0i)T2 1 + (-2.95 + 0.520i)T + (38.5 - 14.0i)T^{2}
43 1+(0.628+7.18i)T+(−42.3+7.46i)T2 1 + (0.628 + 7.18i)T + (-42.3 + 7.46i)T^{2}
47 1+(1.86+0.871i)T+(30.2+36.0i)T2 1 + (1.86 + 0.871i)T + (30.2 + 36.0i)T^{2}
53 1+(1.25−1.25i)T−53iT2 1 + (1.25 - 1.25i)T - 53iT^{2}
59 1+(0.763−0.640i)T+(10.2−58.1i)T2 1 + (0.763 - 0.640i)T + (10.2 - 58.1i)T^{2}
61 1+(6.39−2.32i)T+(46.7−39.2i)T2 1 + (6.39 - 2.32i)T + (46.7 - 39.2i)T^{2}
67 1+(7.45+5.21i)T+(22.9+62.9i)T2 1 + (7.45 + 5.21i)T + (22.9 + 62.9i)T^{2}
71 1+(8.32−4.80i)T+(35.5−61.4i)T2 1 + (8.32 - 4.80i)T + (35.5 - 61.4i)T^{2}
73 1+(2.77−0.744i)T+(63.2−36.5i)T2 1 + (2.77 - 0.744i)T + (63.2 - 36.5i)T^{2}
79 1+(−6.95−1.22i)T+(74.2+27.0i)T2 1 + (-6.95 - 1.22i)T + (74.2 + 27.0i)T^{2}
83 1+(5.84+8.34i)T+(−28.3+77.9i)T2 1 + (5.84 + 8.34i)T + (-28.3 + 77.9i)T^{2}
89 1+(3.03−5.26i)T+(−44.5−77.0i)T2 1 + (3.03 - 5.26i)T + (-44.5 - 77.0i)T^{2}
97 1+(−12.5+1.09i)T+(95.5−16.8i)T2 1 + (-12.5 + 1.09i)T + (95.5 - 16.8i)T^{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

−10.61731803255707141919051188579, −9.663146263014543987397305514360, −8.968087546051583194432931890998, −8.352322773310351476470853519832, −7.57159031002297062721691733612, −6.20948009409687592127225562837, −5.73550220670186939349904729299, −4.21043709353206773076948138447, −2.01289694052453147979383130861, −0.16057895545516383469301059701, 1.81824273636983133504785966789, 3.09956013946915902033109507893, 4.10828301994651221823502962139, 6.25511265304052321855805698158, 7.15390328451871765729485573339, 8.018765185094681621282799405051, 8.962481956267687294265329200994, 10.00522794377417160329755594833, 10.55326523914850883811324216845, 11.06598639092247768997876439105

Graph of the ZZ-function along the critical line