↑

Properties

Label 2-448-1.1-c5-0-1
Degree 22
Conductor 448448
Sign 11
Analytic cond. 71.851971.8519
Root an. cond. 8.476558.47655
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.810·3-s − 71.6·5-s − 49·7-s − 242.·9-s − 569.·11-s − 137.·13-s + 58.0·15-s − 418.·17-s − 2.55e3·19-s + 39.7·21-s − 127.·23-s + 2.01e3·25-s + 393.·27-s − 2.31e3·29-s − 3.99e3·31-s + 461.·33-s + 3.51e3·35-s + 3.85e3·37-s + 111.·39-s − 4.94e3·41-s − 1.36e4·43-s + 1.73e4·45-s + 2.76e4·47-s + 2.40e3·49-s + 339.·51-s − 3.73e4·53-s + 4.08e4·55-s + ⋯
L(s)  = 1  − 0.0519·3-s − 1.28·5-s − 0.377·7-s − 0.997·9-s − 1.41·11-s − 0.225·13-s + 0.0666·15-s − 0.351·17-s − 1.62·19-s + 0.0196·21-s − 0.0500·23-s + 0.643·25-s + 0.103·27-s − 0.510·29-s − 0.746·31-s + 0.0737·33-s + 0.484·35-s + 0.462·37-s + 0.0117·39-s − 0.459·41-s − 1.12·43-s + 1.27·45-s + 1.82·47-s + 0.142·49-s + 0.0182·51-s − 1.82·53-s + 1.81·55-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 11
Analytic conductor: 71.851971.8519
Root analytic conductor: 8.476558.47655
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 448, ( :5/2), 1)(2,\ 448,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.067827691580.06782769158
L(12)L(\frac12) ≈\approx 0.067827691580.06782769158
L(72)L(\frac{7}{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+49T 1 + 49T
good3 1+0.810T+243T2 1 + 0.810T + 243T^{2}
5 1+71.6T+3.12e3T2 1 + 71.6T + 3.12e3T^{2}
11 1+569.T+1.61e5T2 1 + 569.T + 1.61e5T^{2}
13 1+137.T+3.71e5T2 1 + 137.T + 3.71e5T^{2}
17 1+418.T+1.41e6T2 1 + 418.T + 1.41e6T^{2}
19 1+2.55e3T+2.47e6T2 1 + 2.55e3T + 2.47e6T^{2}
23 1+127.T+6.43e6T2 1 + 127.T + 6.43e6T^{2}
29 1+2.31e3T+2.05e7T2 1 + 2.31e3T + 2.05e7T^{2}
31 1+3.99e3T+2.86e7T2 1 + 3.99e3T + 2.86e7T^{2}
37 1−3.85e3T+6.93e7T2 1 - 3.85e3T + 6.93e7T^{2}
41 1+4.94e3T+1.15e8T2 1 + 4.94e3T + 1.15e8T^{2}
43 1+1.36e4T+1.47e8T2 1 + 1.36e4T + 1.47e8T^{2}
47 1−2.76e4T+2.29e8T2 1 - 2.76e4T + 2.29e8T^{2}
53 1+3.73e4T+4.18e8T2 1 + 3.73e4T + 4.18e8T^{2}
59 1+3.69e4T+7.14e8T2 1 + 3.69e4T + 7.14e8T^{2}
61 1−3.80e3T+8.44e8T2 1 - 3.80e3T + 8.44e8T^{2}
67 1−2.24e4T+1.35e9T2 1 - 2.24e4T + 1.35e9T^{2}
71 1−5.50e4T+1.80e9T2 1 - 5.50e4T + 1.80e9T^{2}
73 1+6.92e4T+2.07e9T2 1 + 6.92e4T + 2.07e9T^{2}
79 1−4.09e4T+3.07e9T2 1 - 4.09e4T + 3.07e9T^{2}
83 1−1.97e4T+3.93e9T2 1 - 1.97e4T + 3.93e9T^{2}
89 1−1.04e5T+5.58e9T2 1 - 1.04e5T + 5.58e9T^{2}
97 1+9.66e4T+8.58e9T2 1 + 9.66e4T + 8.58e9T^{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

−10.63478178506018427936318502863, −9.262870250451126872019281470269, −8.276836071218806798378037341099, −7.76674478449369897042364089495, −6.62597626682276292426752736196, −5.50911902101865049701541958350, −4.43773254779062925298312416849, −3.34965852225122933882398643652, −2.32119607809654958738102314697, −0.12399881193292019926013275639, 0.12399881193292019926013275639, 2.32119607809654958738102314697, 3.34965852225122933882398643652, 4.43773254779062925298312416849, 5.50911902101865049701541958350, 6.62597626682276292426752736196, 7.76674478449369897042364089495, 8.276836071218806798378037341099, 9.262870250451126872019281470269, 10.63478178506018427936318502863

Graph of the ZZ-function along the critical line