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Properties

Label 2-405-135.122-c1-0-15
Degree 22
Conductor 405405
Sign 0.387+0.921i0.387 + 0.921i
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  + (1.22 + 0.856i)2-s + (0.0790 + 0.217i)4-s + (−2.10 − 0.751i)5-s + (−2.03 − 4.36i)7-s + (0.683 − 2.55i)8-s + (−1.93 − 2.72i)10-s + (2.25 + 2.68i)11-s + (−2.18 − 3.11i)13-s + (1.24 − 7.08i)14-s + (3.37 − 2.83i)16-s + (0.367 + 1.37i)17-s + (1.30 − 0.750i)19-s + (−0.00328 − 0.517i)20-s + (0.455 + 5.21i)22-s + (−1.35 − 0.633i)23-s + ⋯
L(s)  = 1  + (0.865 + 0.605i)2-s + (0.0395 + 0.108i)4-s + (−0.941 − 0.336i)5-s + (−0.769 − 1.65i)7-s + (0.241 − 0.902i)8-s + (−0.611 − 0.861i)10-s + (0.678 + 0.808i)11-s + (−0.605 − 0.864i)13-s + (0.333 − 1.89i)14-s + (0.844 − 0.708i)16-s + (0.0891 + 0.332i)17-s + (0.298 − 0.172i)19-s + (−0.000733 − 0.115i)20-s + (0.0971 + 1.11i)22-s + (−0.283 − 0.132i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((0.387+0.921i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((0.387+0.921i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: 0.387+0.921i0.387 + 0.921i
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), 0.387+0.921i)(2,\ 405,\ (\ :1/2),\ 0.387 + 0.921i)

Particular Values

L(1)L(1) ≈\approx 1.22309−0.812736i1.22309 - 0.812736i
L(12)L(\frac12) ≈\approx 1.22309−0.812736i1.22309 - 0.812736i
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+(2.10+0.751i)T 1 + (2.10 + 0.751i)T
good2 1+(−1.22−0.856i)T+(0.684+1.87i)T2 1 + (-1.22 - 0.856i)T + (0.684 + 1.87i)T^{2}
7 1+(2.03+4.36i)T+(−4.49+5.36i)T2 1 + (2.03 + 4.36i)T + (-4.49 + 5.36i)T^{2}
11 1+(−2.25−2.68i)T+(−1.91+10.8i)T2 1 + (-2.25 - 2.68i)T + (-1.91 + 10.8i)T^{2}
13 1+(2.18+3.11i)T+(−4.44+12.2i)T2 1 + (2.18 + 3.11i)T + (-4.44 + 12.2i)T^{2}
17 1+(−0.367−1.37i)T+(−14.7+8.5i)T2 1 + (-0.367 - 1.37i)T + (-14.7 + 8.5i)T^{2}
19 1+(−1.30+0.750i)T+(9.5−16.4i)T2 1 + (-1.30 + 0.750i)T + (9.5 - 16.4i)T^{2}
23 1+(1.35+0.633i)T+(14.7+17.6i)T2 1 + (1.35 + 0.633i)T + (14.7 + 17.6i)T^{2}
29 1+(−0.168−0.957i)T+(−27.2+9.91i)T2 1 + (-0.168 - 0.957i)T + (-27.2 + 9.91i)T^{2}
31 1+(2.44−0.891i)T+(23.7−19.9i)T2 1 + (2.44 - 0.891i)T + (23.7 - 19.9i)T^{2}
37 1+(−6.69+1.79i)T+(32.0−18.5i)T2 1 + (-6.69 + 1.79i)T + (32.0 - 18.5i)T^{2}
41 1+(−0.670−0.118i)T+(38.5+14.0i)T2 1 + (-0.670 - 0.118i)T + (38.5 + 14.0i)T^{2}
43 1+(0.0175−0.200i)T+(−42.3−7.46i)T2 1 + (0.0175 - 0.200i)T + (-42.3 - 7.46i)T^{2}
47 1+(−7.89+3.68i)T+(30.2−36.0i)T2 1 + (-7.89 + 3.68i)T + (30.2 - 36.0i)T^{2}
53 1+(2.81+2.81i)T+53iT2 1 + (2.81 + 2.81i)T + 53iT^{2}
59 1+(5.69+4.77i)T+(10.2+58.1i)T2 1 + (5.69 + 4.77i)T + (10.2 + 58.1i)T^{2}
61 1+(1.08+0.396i)T+(46.7+39.2i)T2 1 + (1.08 + 0.396i)T + (46.7 + 39.2i)T^{2}
67 1+(−12.5+8.79i)T+(22.9−62.9i)T2 1 + (-12.5 + 8.79i)T + (22.9 - 62.9i)T^{2}
71 1+(−11.1−6.42i)T+(35.5+61.4i)T2 1 + (-11.1 - 6.42i)T + (35.5 + 61.4i)T^{2}
73 1+(0.343+0.0920i)T+(63.2+36.5i)T2 1 + (0.343 + 0.0920i)T + (63.2 + 36.5i)T^{2}
79 1+(−9.66+1.70i)T+(74.2−27.0i)T2 1 + (-9.66 + 1.70i)T + (74.2 - 27.0i)T^{2}
83 1+(6.51−9.30i)T+(−28.3−77.9i)T2 1 + (6.51 - 9.30i)T + (-28.3 - 77.9i)T^{2}
89 1+(2.09+3.63i)T+(−44.5+77.0i)T2 1 + (2.09 + 3.63i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.41−0.561i)T+(95.5+16.8i)T2 1 + (-6.41 - 0.561i)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

−11.07395326233726167116919003799, −10.14752851046835814630827583984, −9.451557855126933846683664953995, −7.82008889399151131980949153194, −7.22141755773599173949333113491, −6.46104592381420848285295890719, −5.06245763319967509543852978873, −4.16432322838963367412448289431, −3.52542698148374402631186320841, −0.75022747580311212708759917191, 2.43844860885888964601096415526, 3.29151180120175862885977858190, 4.28716593513607995012902271514, 5.51886469229684840178833381389, 6.47380843262574971369420640383, 7.78285837728443059622076771551, 8.781043472240937627163796849220, 9.513368552855250711194179844686, 11.02617996530267894865307876794, 11.77634949237087169474363420572

Graph of the ZZ-function along the critical line