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Properties

Label 2-460-92.91-c1-0-19
Degree 22
Conductor 460460
Sign 0.812−0.583i0.812 - 0.583i
Analytic cond. 3.673113.67311
Root an. cond. 1.916531.91653
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  + (−0.265 + 1.38i)2-s + 0.0612i·3-s + (−1.85 − 0.737i)4-s − i·5-s + (−0.0850 − 0.0162i)6-s − 0.810·7-s + (1.51 − 2.38i)8-s + 2.99·9-s + (1.38 + 0.265i)10-s − 2.10·11-s + (0.0451 − 0.113i)12-s + 2.93·13-s + (0.215 − 1.12i)14-s + 0.0612·15-s + (2.91 + 2.74i)16-s − 4.65i·17-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)2-s + 0.0353i·3-s + (−0.929 − 0.368i)4-s − 0.447i·5-s + (−0.0347 − 0.00663i)6-s − 0.306·7-s + (0.536 − 0.843i)8-s + 0.998·9-s + (0.439 + 0.0838i)10-s − 0.633·11-s + (0.0130 − 0.0328i)12-s + 0.814·13-s + (0.0574 − 0.301i)14-s + 0.0158·15-s + (0.728 + 0.685i)16-s − 1.12i·17-s + ⋯

Functional equation

Λ(s)=(460s/2 ΓC(s) L(s)=((0.812−0.583i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(460s/2 ΓC(s+1/2) L(s)=((0.812−0.583i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 460460    =    22⋅5⋅232^{2} \cdot 5 \cdot 23
Sign: 0.812−0.583i0.812 - 0.583i
Analytic conductor: 3.673113.67311
Root analytic conductor: 1.916531.91653
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ460(91,⋅)\chi_{460} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 460, ( :1/2), 0.812−0.583i)(2,\ 460,\ (\ :1/2),\ 0.812 - 0.583i)

Particular Values

L(1)L(1) ≈\approx 1.17049+0.376503i1.17049 + 0.376503i
L(12)L(\frac12) ≈\approx 1.17049+0.376503i1.17049 + 0.376503i
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)
bad2 1+(0.265−1.38i)T 1 + (0.265 - 1.38i)T
5 1+iT 1 + iT
23 1+(−4.03−2.59i)T 1 + (-4.03 - 2.59i)T
good3 1−0.0612iT−3T2 1 - 0.0612iT - 3T^{2}
7 1+0.810T+7T2 1 + 0.810T + 7T^{2}
11 1+2.10T+11T2 1 + 2.10T + 11T^{2}
13 1−2.93T+13T2 1 - 2.93T + 13T^{2}
17 1+4.65iT−17T2 1 + 4.65iT - 17T^{2}
19 1−6.20T+19T2 1 - 6.20T + 19T^{2}
29 1−5.24T+29T2 1 - 5.24T + 29T^{2}
31 1−2.58iT−31T2 1 - 2.58iT - 31T^{2}
37 1+2.99iT−37T2 1 + 2.99iT - 37T^{2}
41 1−8.21T+41T2 1 - 8.21T + 41T^{2}
43 1+10.1T+43T2 1 + 10.1T + 43T^{2}
47 1−1.89iT−47T2 1 - 1.89iT - 47T^{2}
53 1+8.12iT−53T2 1 + 8.12iT - 53T^{2}
59 1+13.1iT−59T2 1 + 13.1iT - 59T^{2}
61 1−13.7iT−61T2 1 - 13.7iT - 61T^{2}
67 1−1.42T+67T2 1 - 1.42T + 67T^{2}
71 1+2.26iT−71T2 1 + 2.26iT - 71T^{2}
73 1+4.13T+73T2 1 + 4.13T + 73T^{2}
79 1−14.9T+79T2 1 - 14.9T + 79T^{2}
83 1+8.73T+83T2 1 + 8.73T + 83T^{2}
89 1−5.82iT−89T2 1 - 5.82iT - 89T^{2}
97 1−2.73iT−97T2 1 - 2.73iT - 97T^{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

−10.97852896851565101860747736778, −9.847098420096284552363843443167, −9.390527236099553915592277144742, −8.309580702622867806992367748345, −7.42005637937314267802603369159, −6.67691828985568157442388606891, −5.41492611648461168581341565613, −4.72881257974535400125454354572, −3.38091105349331994870739327899, −1.06726062452361544607458848497, 1.30552971756283034367686624868, 2.83713156862426828571034575997, 3.82950017695992429698121672995, 4.96877993887305450983169809771, 6.30669604253359202587465318801, 7.49898395359290323512437948048, 8.365966652616469907780985868809, 9.455300364143450914034286261359, 10.24381445424555626802090763894, 10.80598195217232005446313561769

Graph of the ZZ-function along the critical line