↑

Properties

Label 2-448-1.1-c3-0-17
Degree 22
Conductor 448448
Sign 11
Analytic cond. 26.432826.4328
Root an. cond. 5.141285.14128
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.64·3-s + 18.3·5-s − 7·7-s − 5.38·9-s + 39.5·11-s − 64.5·13-s + 85.2·15-s + 109.·17-s + 137.·19-s − 32.5·21-s − 45.2·23-s + 211.·25-s − 150.·27-s + 41.1·29-s + 262.·31-s + 184·33-s − 128.·35-s − 125.·37-s − 299.·39-s − 299.·41-s + 36.9·43-s − 98.7·45-s + 122.·47-s + 49·49-s + 507.·51-s + 20.4·53-s + 725.·55-s + ⋯
L(s)  = 1  + 0.894·3-s + 1.63·5-s − 0.377·7-s − 0.199·9-s + 1.08·11-s − 1.37·13-s + 1.46·15-s + 1.55·17-s + 1.65·19-s − 0.338·21-s − 0.410·23-s + 1.68·25-s − 1.07·27-s + 0.263·29-s + 1.52·31-s + 0.970·33-s − 0.619·35-s − 0.558·37-s − 1.23·39-s − 1.14·41-s + 0.130·43-s − 0.327·45-s + 0.381·47-s + 0.142·49-s + 1.39·51-s + 0.0530·53-s + 1.77·55-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 11
Analytic conductor: 26.432826.4328
Root analytic conductor: 5.141285.14128
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 448, ( :3/2), 1)(2,\ 448,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.5343959643.534395964
L(12)L(\frac12) ≈\approx 3.5343959643.534395964
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
7 1+7T 1 + 7T
good3 1−4.64T+27T2 1 - 4.64T + 27T^{2}
5 1−18.3T+125T2 1 - 18.3T + 125T^{2}
11 1−39.5T+1.33e3T2 1 - 39.5T + 1.33e3T^{2}
13 1+64.5T+2.19e3T2 1 + 64.5T + 2.19e3T^{2}
17 1−109.T+4.91e3T2 1 - 109.T + 4.91e3T^{2}
19 1−137.T+6.85e3T2 1 - 137.T + 6.85e3T^{2}
23 1+45.2T+1.21e4T2 1 + 45.2T + 1.21e4T^{2}
29 1−41.1T+2.43e4T2 1 - 41.1T + 2.43e4T^{2}
31 1−262.T+2.97e4T2 1 - 262.T + 2.97e4T^{2}
37 1+125.T+5.06e4T2 1 + 125.T + 5.06e4T^{2}
41 1+299.T+6.89e4T2 1 + 299.T + 6.89e4T^{2}
43 1−36.9T+7.95e4T2 1 - 36.9T + 7.95e4T^{2}
47 1−122.T+1.03e5T2 1 - 122.T + 1.03e5T^{2}
53 1−20.4T+1.48e5T2 1 - 20.4T + 1.48e5T^{2}
59 1+60.8T+2.05e5T2 1 + 60.8T + 2.05e5T^{2}
61 1+791.T+2.26e5T2 1 + 791.T + 2.26e5T^{2}
67 1−1.04e3T+3.00e5T2 1 - 1.04e3T + 3.00e5T^{2}
71 1+407.T+3.57e5T2 1 + 407.T + 3.57e5T^{2}
73 1−562.T+3.89e5T2 1 - 562.T + 3.89e5T^{2}
79 1−601.T+4.93e5T2 1 - 601.T + 4.93e5T^{2}
83 1+652.T+5.71e5T2 1 + 652.T + 5.71e5T^{2}
89 1+898.T+7.04e5T2 1 + 898.T + 7.04e5T^{2}
97 1+621.T+9.12e5T2 1 + 621.T + 9.12e5T^{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

−10.12403866012600281130442891663, −9.728917805532807568966406538560, −9.180652938648264341081899114033, −8.025819700228228576250898279123, −6.96728386966855304842691162848, −5.91901564828315865149073237603, −5.09299878756849307513215298212, −3.37461920163295047860819015647, −2.53506010118222026204505841790, −1.29925975829022415025363740226, 1.29925975829022415025363740226, 2.53506010118222026204505841790, 3.37461920163295047860819015647, 5.09299878756849307513215298212, 5.91901564828315865149073237603, 6.96728386966855304842691162848, 8.025819700228228576250898279123, 9.180652938648264341081899114033, 9.728917805532807568966406538560, 10.12403866012600281130442891663

Graph of the ZZ-function along the critical line