↑

Properties

Label 2-416-416.77-c1-0-11
Degree 22
Conductor 416416
Sign 0.580−0.814i0.580 - 0.814i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 + 0.510i)2-s + (0.324 − 0.782i)3-s + (1.47 − 1.34i)4-s + (0.174 + 0.420i)5-s + (−0.0279 + 1.19i)6-s + (2.63 + 2.63i)7-s + (−1.26 + 2.53i)8-s + (1.61 + 1.61i)9-s + (−0.444 − 0.465i)10-s + (−2.03 + 0.842i)11-s + (−0.574 − 1.59i)12-s + (−2.76 − 2.31i)13-s + (−4.81 − 2.12i)14-s + 0.385·15-s + (0.373 − 3.98i)16-s + 4.77i·17-s + ⋯
L(s)  = 1  + (−0.932 + 0.360i)2-s + (0.187 − 0.451i)3-s + (0.739 − 0.673i)4-s + (0.0778 + 0.187i)5-s + (−0.0114 + 0.488i)6-s + (0.995 + 0.995i)7-s + (−0.446 + 0.894i)8-s + (0.538 + 0.538i)9-s + (−0.140 − 0.147i)10-s + (−0.613 + 0.254i)11-s + (−0.165 − 0.459i)12-s + (−0.765 − 0.643i)13-s + (−1.28 − 0.568i)14-s + 0.0994·15-s + (0.0933 − 0.995i)16-s + 1.15i·17-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.580−0.814i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.580−0.814i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.580−0.814i0.580 - 0.814i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(77,⋅)\chi_{416} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.580−0.814i)(2,\ 416,\ (\ :1/2),\ 0.580 - 0.814i)

Particular Values

L(1)L(1) ≈\approx 0.932958+0.480560i0.932958 + 0.480560i
L(12)L(\frac12) ≈\approx 0.932958+0.480560i0.932958 + 0.480560i
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+(1.31−0.510i)T 1 + (1.31 - 0.510i)T
13 1+(2.76+2.31i)T 1 + (2.76 + 2.31i)T
good3 1+(−0.324+0.782i)T+(−2.12−2.12i)T2 1 + (-0.324 + 0.782i)T + (-2.12 - 2.12i)T^{2}
5 1+(−0.174−0.420i)T+(−3.53+3.53i)T2 1 + (-0.174 - 0.420i)T + (-3.53 + 3.53i)T^{2}
7 1+(−2.63−2.63i)T+7iT2 1 + (-2.63 - 2.63i)T + 7iT^{2}
11 1+(2.03−0.842i)T+(7.77−7.77i)T2 1 + (2.03 - 0.842i)T + (7.77 - 7.77i)T^{2}
17 1−4.77iT−17T2 1 - 4.77iT - 17T^{2}
19 1+(0.598−1.44i)T+(−13.4−13.4i)T2 1 + (0.598 - 1.44i)T + (-13.4 - 13.4i)T^{2}
23 1+(−3.12−3.12i)T+23iT2 1 + (-3.12 - 3.12i)T + 23iT^{2}
29 1+(1.02−2.47i)T+(−20.5−20.5i)T2 1 + (1.02 - 2.47i)T + (-20.5 - 20.5i)T^{2}
31 1−1.90iT−31T2 1 - 1.90iT - 31T^{2}
37 1+(2.32+5.61i)T+(−26.1+26.1i)T2 1 + (2.32 + 5.61i)T + (-26.1 + 26.1i)T^{2}
41 1+(−2.95+2.95i)T−41iT2 1 + (-2.95 + 2.95i)T - 41iT^{2}
43 1+(−4.00−9.66i)T+(−30.4+30.4i)T2 1 + (-4.00 - 9.66i)T + (-30.4 + 30.4i)T^{2}
47 1−1.36T+47T2 1 - 1.36T + 47T^{2}
53 1+(0.464+1.12i)T+(−37.4+37.4i)T2 1 + (0.464 + 1.12i)T + (-37.4 + 37.4i)T^{2}
59 1+(−0.498−1.20i)T+(−41.7+41.7i)T2 1 + (-0.498 - 1.20i)T + (-41.7 + 41.7i)T^{2}
61 1+(−3.55+8.57i)T+(−43.1−43.1i)T2 1 + (-3.55 + 8.57i)T + (-43.1 - 43.1i)T^{2}
67 1+(−7.70−3.19i)T+(47.3+47.3i)T2 1 + (-7.70 - 3.19i)T + (47.3 + 47.3i)T^{2}
71 1+(9.25+9.25i)T+71iT2 1 + (9.25 + 9.25i)T + 71iT^{2}
73 1+(−10.4+10.4i)T−73iT2 1 + (-10.4 + 10.4i)T - 73iT^{2}
79 1+1.42iT−79T2 1 + 1.42iT - 79T^{2}
83 1+(4.61−11.1i)T+(−58.6−58.6i)T2 1 + (4.61 - 11.1i)T + (-58.6 - 58.6i)T^{2}
89 1+(5.78+5.78i)T+89iT2 1 + (5.78 + 5.78i)T + 89iT^{2}
97 1+9.17iT−97T2 1 + 9.17iT - 97T^{2}
show more
show less
   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

−10.98400193997475920133298165407, −10.50095928790751261219471866027, −9.422933151520359080559142677395, −8.330529324677037419726373152451, −7.87673964788035726621938233810, −6.98324414203059736353295124834, −5.71603885271011729772341609745, −4.90712649601190569219547259059, −2.60115287358147544115837895481, −1.67500491566344341712099176199, 0.976107232010067268318660425407, 2.61623853066820194243635591961, 4.05793620258765451436399181766, 4.99768094966376238468077017643, 6.90254466857108773226776792193, 7.39641088510646886575445938060, 8.519111416202564210592424520651, 9.351796731429093648189992587138, 10.12209437912093059011656228234, 10.92095638208681322079427104263

Graph of the ZZ-function along the critical line