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Properties

Label 2-448-1.1-c5-0-11
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.9·3-s − 106.·5-s + 49·7-s + 156.·9-s − 452.·11-s − 886.·13-s − 2.12e3·15-s + 297.·17-s + 2.28e3·19-s + 978.·21-s − 555.·23-s + 8.14e3·25-s − 1.73e3·27-s + 8.26e3·29-s − 4.24e3·31-s − 9.03e3·33-s − 5.20e3·35-s + 758.·37-s − 1.77e4·39-s + 1.72e4·41-s − 5.37e3·43-s − 1.65e4·45-s + 2.56e4·47-s + 2.40e3·49-s + 5.95e3·51-s − 1.08e4·53-s + 4.80e4·55-s + ⋯
L(s)  = 1  + 1.28·3-s − 1.89·5-s + 0.377·7-s + 0.642·9-s − 1.12·11-s − 1.45·13-s − 2.43·15-s + 0.250·17-s + 1.45·19-s + 0.484·21-s − 0.218·23-s + 2.60·25-s − 0.458·27-s + 1.82·29-s − 0.793·31-s − 1.44·33-s − 0.717·35-s + 0.0911·37-s − 1.86·39-s + 1.59·41-s − 0.443·43-s − 1.22·45-s + 1.69·47-s + 0.142·49-s + 0.320·51-s − 0.529·53-s + 2.14·55-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 1.8179788841.817978884
L(12)L(\frac12) ≈\approx 1.8179788841.817978884
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 1−19.9T+243T2 1 - 19.9T + 243T^{2}
5 1+106.T+3.12e3T2 1 + 106.T + 3.12e3T^{2}
11 1+452.T+1.61e5T2 1 + 452.T + 1.61e5T^{2}
13 1+886.T+3.71e5T2 1 + 886.T + 3.71e5T^{2}
17 1−297.T+1.41e6T2 1 - 297.T + 1.41e6T^{2}
19 1−2.28e3T+2.47e6T2 1 - 2.28e3T + 2.47e6T^{2}
23 1+555.T+6.43e6T2 1 + 555.T + 6.43e6T^{2}
29 1−8.26e3T+2.05e7T2 1 - 8.26e3T + 2.05e7T^{2}
31 1+4.24e3T+2.86e7T2 1 + 4.24e3T + 2.86e7T^{2}
37 1−758.T+6.93e7T2 1 - 758.T + 6.93e7T^{2}
41 1−1.72e4T+1.15e8T2 1 - 1.72e4T + 1.15e8T^{2}
43 1+5.37e3T+1.47e8T2 1 + 5.37e3T + 1.47e8T^{2}
47 1−2.56e4T+2.29e8T2 1 - 2.56e4T + 2.29e8T^{2}
53 1+1.08e4T+4.18e8T2 1 + 1.08e4T + 4.18e8T^{2}
59 1+2.79e3T+7.14e8T2 1 + 2.79e3T + 7.14e8T^{2}
61 1+8.46e3T+8.44e8T2 1 + 8.46e3T + 8.44e8T^{2}
67 1−1.43e4T+1.35e9T2 1 - 1.43e4T + 1.35e9T^{2}
71 1−6.11e4T+1.80e9T2 1 - 6.11e4T + 1.80e9T^{2}
73 1+6.00e3T+2.07e9T2 1 + 6.00e3T + 2.07e9T^{2}
79 1+2.53e4T+3.07e9T2 1 + 2.53e4T + 3.07e9T^{2}
83 1+5.43e3T+3.93e9T2 1 + 5.43e3T + 3.93e9T^{2}
89 1−3.03e4T+5.58e9T2 1 - 3.03e4T + 5.58e9T^{2}
97 1+8.30e4T+8.58e9T2 1 + 8.30e4T + 8.58e9T^{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

−10.23096957490264221250857643221, −9.184411434767617985481926968496, −8.205292313340335857147824730231, −7.68265108143760142150628989524, −7.26635396881615425826118859107, −5.19871357165099204775188754814, −4.30396408458670992657539385461, −3.21075401051951307698509459443, −2.54130234799171958191716627889, −0.63242666027610530723386839012, 0.63242666027610530723386839012, 2.54130234799171958191716627889, 3.21075401051951307698509459443, 4.30396408458670992657539385461, 5.19871357165099204775188754814, 7.26635396881615425826118859107, 7.68265108143760142150628989524, 8.205292313340335857147824730231, 9.184411434767617985481926968496, 10.23096957490264221250857643221

Graph of the ZZ-function along the critical line