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Properties

Label 2-460-92.91-c1-0-14
Degree 22
Conductor 460460
Sign −0.969−0.246i-0.969 - 0.246i
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  + (0.949 + 1.04i)2-s + 2.47i·3-s + (−0.198 + 1.99i)4-s + i·5-s + (−2.59 + 2.35i)6-s + 1.26·7-s + (−2.27 + 1.68i)8-s − 3.14·9-s + (−1.04 + 0.949i)10-s + 3.32·11-s + (−4.93 − 0.491i)12-s + 1.65·13-s + (1.19 + 1.32i)14-s − 2.47·15-s + (−3.92 − 0.788i)16-s − 6.70i·17-s + ⋯
L(s)  = 1  + (0.671 + 0.741i)2-s + 1.43i·3-s + (−0.0990 + 0.995i)4-s + 0.447i·5-s + (−1.06 + 0.960i)6-s + 0.476·7-s + (−0.804 + 0.594i)8-s − 1.04·9-s + (−0.331 + 0.300i)10-s + 1.00·11-s + (−1.42 − 0.141i)12-s + 0.457·13-s + (0.319 + 0.353i)14-s − 0.640·15-s + (−0.980 − 0.197i)16-s − 1.62i·17-s + ⋯

Functional equation

Λ(s)=(460s/2 ΓC(s) L(s)=((−0.969−0.246i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(460s/2 ΓC(s+1/2) L(s)=((−0.969−0.246i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 460460    =    22⋅5⋅232^{2} \cdot 5 \cdot 23
Sign: −0.969−0.246i-0.969 - 0.246i
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), −0.969−0.246i)(2,\ 460,\ (\ :1/2),\ -0.969 - 0.246i)

Particular Values

L(1)L(1) ≈\approx 0.254734+2.03842i0.254734 + 2.03842i
L(12)L(\frac12) ≈\approx 0.254734+2.03842i0.254734 + 2.03842i
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+(−0.949−1.04i)T 1 + (-0.949 - 1.04i)T
5 1−iT 1 - iT
23 1+(4.74+0.713i)T 1 + (4.74 + 0.713i)T
good3 1−2.47iT−3T2 1 - 2.47iT - 3T^{2}
7 1−1.26T+7T2 1 - 1.26T + 7T^{2}
11 1−3.32T+11T2 1 - 3.32T + 11T^{2}
13 1−1.65T+13T2 1 - 1.65T + 13T^{2}
17 1+6.70iT−17T2 1 + 6.70iT - 17T^{2}
19 1+0.390T+19T2 1 + 0.390T + 19T^{2}
29 1−8.11T+29T2 1 - 8.11T + 29T^{2}
31 1+2.86iT−31T2 1 + 2.86iT - 31T^{2}
37 1+6.44iT−37T2 1 + 6.44iT - 37T^{2}
41 1−4.47T+41T2 1 - 4.47T + 41T^{2}
43 1+2.71T+43T2 1 + 2.71T + 43T^{2}
47 1−0.827iT−47T2 1 - 0.827iT - 47T^{2}
53 1−8.17iT−53T2 1 - 8.17iT - 53T^{2}
59 1−4.19iT−59T2 1 - 4.19iT - 59T^{2}
61 1−5.01iT−61T2 1 - 5.01iT - 61T^{2}
67 1+13.4T+67T2 1 + 13.4T + 67T^{2}
71 1−9.80iT−71T2 1 - 9.80iT - 71T^{2}
73 1−3.36T+73T2 1 - 3.36T + 73T^{2}
79 1−15.5T+79T2 1 - 15.5T + 79T^{2}
83 1−1.10T+83T2 1 - 1.10T + 83T^{2}
89 1+8.50iT−89T2 1 + 8.50iT - 89T^{2}
97 1−7.78iT−97T2 1 - 7.78iT - 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

−11.51762046300091376529510066335, −10.62432016410925530609697558543, −9.527892195925020304112093939295, −8.872210088116512579196447448860, −7.75928606504816423893391500552, −6.67364159188318529685338839709, −5.67601455816756861865105053983, −4.60766616106242609368552561320, −3.99440469734202914129136381235, −2.84762487184491092627172563837, 1.20666264276522374159147871729, 1.94623946377542538174025954506, 3.60725988312691672988162563631, 4.74319404292150782182713211059, 6.15905600231019505009167315517, 6.51610941970317339312413179116, 8.014662048207496638068723516831, 8.673376197795191016703137430684, 9.938893633646080026423997466832, 10.96279781143286395477831190244

Graph of the ZZ-function along the critical line