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Properties

Label 2-460-92.91-c1-0-43
Degree 22
Conductor 460460
Sign −0.574−0.818i-0.574 - 0.818i
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  + (−0.151 − 1.40i)2-s − 2.99i·3-s + (−1.95 + 0.425i)4-s − i·5-s + (−4.21 + 0.453i)6-s + 3.98·7-s + (0.893 + 2.68i)8-s − 5.98·9-s + (−1.40 + 0.151i)10-s − 5.79·11-s + (1.27 + 5.85i)12-s − 5.59·13-s + (−0.602 − 5.59i)14-s − 2.99·15-s + (3.63 − 1.66i)16-s − 1.88i·17-s + ⋯
L(s)  = 1  + (−0.106 − 0.994i)2-s − 1.73i·3-s + (−0.977 + 0.212i)4-s − 0.447i·5-s + (−1.72 + 0.185i)6-s + 1.50·7-s + (0.315 + 0.948i)8-s − 1.99·9-s + (−0.444 + 0.0478i)10-s − 1.74·11-s + (0.368 + 1.69i)12-s − 1.55·13-s + (−0.160 − 1.49i)14-s − 0.774·15-s + (0.909 − 0.415i)16-s − 0.456i·17-s + ⋯

Functional equation

Λ(s)=(460s/2 ΓC(s) L(s)=((−0.574−0.818i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(460s/2 ΓC(s+1/2) L(s)=((−0.574−0.818i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 460460    =    22⋅5⋅232^{2} \cdot 5 \cdot 23
Sign: −0.574−0.818i-0.574 - 0.818i
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), −0.574−0.818i)(2,\ 460,\ (\ :1/2),\ -0.574 - 0.818i)

Particular Values

L(1)L(1) ≈\approx 0.441970+0.850728i0.441970 + 0.850728i
L(12)L(\frac12) ≈\approx 0.441970+0.850728i0.441970 + 0.850728i
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+(0.151+1.40i)T 1 + (0.151 + 1.40i)T
5 1+iT 1 + iT
23 1+(−4.42+1.85i)T 1 + (-4.42 + 1.85i)T
good3 1+2.99iT−3T2 1 + 2.99iT - 3T^{2}
7 1−3.98T+7T2 1 - 3.98T + 7T^{2}
11 1+5.79T+11T2 1 + 5.79T + 11T^{2}
13 1+5.59T+13T2 1 + 5.59T + 13T^{2}
17 1+1.88iT−17T2 1 + 1.88iT - 17T^{2}
19 1−3.03T+19T2 1 - 3.03T + 19T^{2}
29 1−1.45T+29T2 1 - 1.45T + 29T^{2}
31 1+4.43iT−31T2 1 + 4.43iT - 31T^{2}
37 1−4.01iT−37T2 1 - 4.01iT - 37T^{2}
41 1+2.68T+41T2 1 + 2.68T + 41T^{2}
43 1−4.78T+43T2 1 - 4.78T + 43T^{2}
47 1+6.45iT−47T2 1 + 6.45iT - 47T^{2}
53 1+8.62iT−53T2 1 + 8.62iT - 53T^{2}
59 1+1.55iT−59T2 1 + 1.55iT - 59T^{2}
61 1+6.06iT−61T2 1 + 6.06iT - 61T^{2}
67 1−7.88T+67T2 1 - 7.88T + 67T^{2}
71 1+5.26iT−71T2 1 + 5.26iT - 71T^{2}
73 1−2.20T+73T2 1 - 2.20T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1+8.56T+83T2 1 + 8.56T + 83T^{2}
89 1−8.41iT−89T2 1 - 8.41iT - 89T^{2}
97 1−2.87iT−97T2 1 - 2.87iT - 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

−10.78954446804749818840699349094, −9.621551956258333915691909472452, −8.311876130051242789514409996929, −7.929457395327154634284168809856, −7.19768456071297107840759877993, −5.25483354048829077739661550086, −4.96886432780788319119972028431, −2.70009590510981186090186997322, −1.99565578732841433138515620392, −0.61214061972460443618340604014, 2.85512985426219632672877783302, 4.35438417574474404048069356418, 5.10999497199785456639401690401, 5.47510199844017187060771949450, 7.34462840742329801187190302124, 7.971609101245998314781228495460, 8.941614086313297277975269415817, 9.884112496296329222903266765096, 10.52997501818797086249122821872, 11.15746801067043255971326232320

Graph of the ZZ-function along the critical line