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Properties

Label 2-448-8.5-c3-0-4
Degree 22
Conductor 448448
Sign 0.965−0.258i0.965 - 0.258i
Analytic cond. 26.432826.4328
Root an. cond. 5.141285.14128
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.66i·3-s + 11.8i·5-s − 7·7-s − 66.3·9-s + 36.1i·11-s − 59.5i·13-s + 114.·15-s − 28.4·17-s + 128. i·19-s + 67.6i·21-s + 168.·23-s − 14.3·25-s + 379. i·27-s + 24.8i·29-s + 98.4·31-s + ⋯
L(s)  = 1  − 1.85i·3-s + 1.05i·5-s − 0.377·7-s − 2.45·9-s + 0.989i·11-s − 1.26i·13-s + 1.96·15-s − 0.405·17-s + 1.54i·19-s + 0.702i·21-s + 1.52·23-s − 0.114·25-s + 2.70i·27-s + 0.159i·29-s + 0.570·31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((0.965−0.258i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=((0.965−0.258i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 0.965−0.258i0.965 - 0.258i
Analytic conductor: 26.432826.4328
Root analytic conductor: 5.141285.14128
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ448(225,⋅)\chi_{448} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 448, ( :3/2), 0.965−0.258i)(2,\ 448,\ (\ :3/2),\ 0.965 - 0.258i)

Particular Values

L(2)L(2) ≈\approx 1.3100213251.310021325
L(12)L(\frac12) ≈\approx 1.3100213251.310021325
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+9.66iT−27T2 1 + 9.66iT - 27T^{2}
5 1−11.8iT−125T2 1 - 11.8iT - 125T^{2}
11 1−36.1iT−1.33e3T2 1 - 36.1iT - 1.33e3T^{2}
13 1+59.5iT−2.19e3T2 1 + 59.5iT - 2.19e3T^{2}
17 1+28.4T+4.91e3T2 1 + 28.4T + 4.91e3T^{2}
19 1−128.iT−6.85e3T2 1 - 128. iT - 6.85e3T^{2}
23 1−168.T+1.21e4T2 1 - 168.T + 1.21e4T^{2}
29 1−24.8iT−2.43e4T2 1 - 24.8iT - 2.43e4T^{2}
31 1−98.4T+2.97e4T2 1 - 98.4T + 2.97e4T^{2}
37 1−411.iT−5.06e4T2 1 - 411. iT - 5.06e4T^{2}
41 1+302.T+6.89e4T2 1 + 302.T + 6.89e4T^{2}
43 1+61.8iT−7.95e4T2 1 + 61.8iT - 7.95e4T^{2}
47 1−493.T+1.03e5T2 1 - 493.T + 1.03e5T^{2}
53 1+66.6iT−1.48e5T2 1 + 66.6iT - 1.48e5T^{2}
59 1+467.iT−2.05e5T2 1 + 467. iT - 2.05e5T^{2}
61 1−315.iT−2.26e5T2 1 - 315. iT - 2.26e5T^{2}
67 1−765.iT−3.00e5T2 1 - 765. iT - 3.00e5T^{2}
71 1−241.T+3.57e5T2 1 - 241.T + 3.57e5T^{2}
73 1−413.T+3.89e5T2 1 - 413.T + 3.89e5T^{2}
79 1−711.T+4.93e5T2 1 - 711.T + 4.93e5T^{2}
83 1+147.iT−5.71e5T2 1 + 147. iT - 5.71e5T^{2}
89 1+425.T+7.04e5T2 1 + 425.T + 7.04e5T^{2}
97 1+1.13e3T+9.12e5T2 1 + 1.13e3T + 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

−10.79462735586476107754906849814, −9.998967726309375730302891819541, −8.542251614489064108934395022037, −7.72957646232937544055915110895, −6.95570892124293237947612448222, −6.43445186697888168894355703773, −5.33888251068189840874242508825, −3.26882376490822634159865755793, −2.44200337852162393570086843706, −1.14253313550994790905638192966, 0.47808367271363901489468562474, 2.79549638213313871063582555833, 3.97941457016640019111177580940, 4.74793187066454165833480389282, 5.48161204123068506809442082323, 6.74859142880018561343366325730, 8.505579671419432453734860205829, 9.179505070820597702729529166442, 9.332233631117183444233425624763, 10.79634026873518485732861044020

Graph of the ZZ-function along the critical line