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Properties

Label 2-2e8-1.1-c3-0-6
Degree 22
Conductor 256256
Sign 11
Analytic cond. 15.104415.1044
Root an. cond. 3.886443.88644
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.32·3-s − 17.8·5-s + 22.6·7-s + 13.0·9-s + 44.2·11-s − 17.8·13-s − 113.·15-s + 70·17-s + 82.2·19-s + 143.·21-s + 158.·23-s + 195.·25-s − 88.5·27-s + 125.·29-s + 280·33-s − 404.·35-s − 375.·37-s − 113.·39-s + 182·41-s − 132.·43-s − 232.·45-s + 316.·47-s + 169.·49-s + 442.·51-s + 125.·53-s − 791.·55-s + 520·57-s + ⋯
L(s)  = 1  + 1.21·3-s − 1.60·5-s + 1.22·7-s + 0.481·9-s + 1.21·11-s − 0.381·13-s − 1.94·15-s + 0.998·17-s + 0.992·19-s + 1.48·21-s + 1.43·23-s + 1.56·25-s − 0.631·27-s + 0.801·29-s + 1.47·33-s − 1.95·35-s − 1.66·37-s − 0.464·39-s + 0.693·41-s − 0.471·43-s − 0.770·45-s + 0.983·47-s + 0.492·49-s + 1.21·51-s + 0.324·53-s − 1.94·55-s + 1.20·57-s + ⋯

Functional equation

Λ(s)=(256s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(256s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 256256    =    282^{8}
Sign: 11
Analytic conductor: 15.104415.1044
Root analytic conductor: 3.886443.88644
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 256, ( :3/2), 1)(2,\ 256,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.5263100592.526310059
L(12)L(\frac12) ≈\approx 2.5263100592.526310059
L(52)L(\frac{5}{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
good3 1−6.32T+27T2 1 - 6.32T + 27T^{2}
5 1+17.8T+125T2 1 + 17.8T + 125T^{2}
7 1−22.6T+343T2 1 - 22.6T + 343T^{2}
11 1−44.2T+1.33e3T2 1 - 44.2T + 1.33e3T^{2}
13 1+17.8T+2.19e3T2 1 + 17.8T + 2.19e3T^{2}
17 1−70T+4.91e3T2 1 - 70T + 4.91e3T^{2}
19 1−82.2T+6.85e3T2 1 - 82.2T + 6.85e3T^{2}
23 1−158.T+1.21e4T2 1 - 158.T + 1.21e4T^{2}
29 1−125.T+2.43e4T2 1 - 125.T + 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1+375.T+5.06e4T2 1 + 375.T + 5.06e4T^{2}
41 1−182T+6.89e4T2 1 - 182T + 6.89e4T^{2}
43 1+132.T+7.95e4T2 1 + 132.T + 7.95e4T^{2}
47 1−316.T+1.03e5T2 1 - 316.T + 1.03e5T^{2}
53 1−125.T+1.48e5T2 1 - 125.T + 1.48e5T^{2}
59 1+82.2T+2.05e5T2 1 + 82.2T + 2.05e5T^{2}
61 1+232.T+2.26e5T2 1 + 232.T + 2.26e5T^{2}
67 1+221.T+3.00e5T2 1 + 221.T + 3.00e5T^{2}
71 1−113.T+3.57e5T2 1 - 113.T + 3.57e5T^{2}
73 1+910T+3.89e5T2 1 + 910T + 3.89e5T^{2}
79 1−678.T+4.93e5T2 1 - 678.T + 4.93e5T^{2}
83 1−714.T+5.71e5T2 1 - 714.T + 5.71e5T^{2}
89 1−546T+7.04e5T2 1 - 546T + 7.04e5T^{2}
97 1+490T+9.12e5T2 1 + 490T + 9.12e5T^{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

−11.79241142976214084938224259015, −10.78053927458870975516483751194, −9.285986175938985519340728425500, −8.532196739042168954869126170542, −7.77110021882884065442369727389, −7.12573394853413171373022686224, −5.06133754337830436455469454741, −3.94581939992963004162318102018, −3.06148260417604879894659198697, −1.22451930226900963150378655118, 1.22451930226900963150378655118, 3.06148260417604879894659198697, 3.94581939992963004162318102018, 5.06133754337830436455469454741, 7.12573394853413171373022686224, 7.77110021882884065442369727389, 8.532196739042168954869126170542, 9.285986175938985519340728425500, 10.78053927458870975516483751194, 11.79241142976214084938224259015

Graph of the ZZ-function along the critical line