↑

Properties

Label 2-448-1.1-c5-0-17
Degree 22
Conductor 448448
Sign 11
Analytic cond. 71.851971.8519
Root an. cond. 8.476558.47655
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 96·5-s − 49·7-s − 239·9-s − 720·11-s − 572·13-s − 192·15-s + 1.25e3·17-s − 94·19-s + 98·21-s − 96·23-s + 6.09e3·25-s + 964·27-s + 4.37e3·29-s + 6.24e3·31-s + 1.44e3·33-s − 4.70e3·35-s + 1.07e4·37-s + 1.14e3·39-s + 1.20e4·41-s − 9.16e3·43-s − 2.29e4·45-s + 2.58e4·47-s + 2.40e3·49-s − 2.50e3·51-s − 1.01e3·53-s − 6.91e4·55-s + ⋯
L(s)  = 1  − 0.128·3-s + 1.71·5-s − 0.377·7-s − 0.983·9-s − 1.79·11-s − 0.938·13-s − 0.220·15-s + 1.05·17-s − 0.0597·19-s + 0.0484·21-s − 0.0378·23-s + 1.94·25-s + 0.254·27-s + 0.965·29-s + 1.16·31-s + 0.230·33-s − 0.649·35-s + 1.29·37-s + 0.120·39-s + 1.11·41-s − 0.755·43-s − 1.68·45-s + 1.70·47-s + 1/7·49-s − 0.135·51-s − 0.0495·53-s − 3.08·55-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: yes
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 2.1187765422.118776542
L(12)L(\frac12) ≈\approx 2.1187765422.118776542
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+p2T 1 + p^{2} T
good3 1+2T+p5T2 1 + 2 T + p^{5} T^{2}
5 1−96T+p5T2 1 - 96 T + p^{5} T^{2}
11 1+720T+p5T2 1 + 720 T + p^{5} T^{2}
13 1+44pT+p5T2 1 + 44 p T + p^{5} T^{2}
17 1−1254T+p5T2 1 - 1254 T + p^{5} T^{2}
19 1+94T+p5T2 1 + 94 T + p^{5} T^{2}
23 1+96T+p5T2 1 + 96 T + p^{5} T^{2}
29 1−4374T+p5T2 1 - 4374 T + p^{5} T^{2}
31 1−6244T+p5T2 1 - 6244 T + p^{5} T^{2}
37 1−10798T+p5T2 1 - 10798 T + p^{5} T^{2}
41 1−12006T+p5T2 1 - 12006 T + p^{5} T^{2}
43 1+9160T+p5T2 1 + 9160 T + p^{5} T^{2}
47 1−25836T+p5T2 1 - 25836 T + p^{5} T^{2}
53 1+1014T+p5T2 1 + 1014 T + p^{5} T^{2}
59 1−1242T+p5T2 1 - 1242 T + p^{5} T^{2}
61 1+7592T+p5T2 1 + 7592 T + p^{5} T^{2}
67 1−41132T+p5T2 1 - 41132 T + p^{5} T^{2}
71 1−37632T+p5T2 1 - 37632 T + p^{5} T^{2}
73 1+13438T+p5T2 1 + 13438 T + p^{5} T^{2}
79 1+6248T+p5T2 1 + 6248 T + p^{5} T^{2}
83 1+25254T+p5T2 1 + 25254 T + p^{5} T^{2}
89 1+45126T+p5T2 1 + 45126 T + p^{5} T^{2}
97 1−107222T+p5T2 1 - 107222 T + p^{5} T^{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

−10.08278606912181803038844356056, −9.722863581771719520871478508012, −8.535719485284051150169392476463, −7.56997175019110947800037908400, −6.26310282912933770517725114252, −5.60012416596259633608302326142, −4.91504362725559381700277952829, −2.79684580863203649306946847621, −2.45057591932716256621523526145, −0.73170272917744731999750165833, 0.73170272917744731999750165833, 2.45057591932716256621523526145, 2.79684580863203649306946847621, 4.91504362725559381700277952829, 5.60012416596259633608302326142, 6.26310282912933770517725114252, 7.56997175019110947800037908400, 8.535719485284051150169392476463, 9.722863581771719520871478508012, 10.08278606912181803038844356056

Graph of the ZZ-function along the critical line