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Properties

Label 2-448-112.27-c3-0-13
Degree 22
Conductor 448448
Sign −0.612−0.790i-0.612 - 0.790i
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  + (−7.12 + 7.12i)3-s + (5.44 − 5.44i)5-s + (−16.6 − 8.05i)7-s − 74.5i·9-s + (12.2 − 12.2i)11-s + (14.9 + 14.9i)13-s + 77.5i·15-s + 97.2i·17-s + (65.2 − 65.2i)19-s + (176. − 61.4i)21-s − 51.7·23-s + 65.7i·25-s + (338. + 338. i)27-s + (19.0 − 19.0i)29-s + 112.·31-s + ⋯
L(s)  = 1  + (−1.37 + 1.37i)3-s + (0.486 − 0.486i)5-s + (−0.900 − 0.435i)7-s − 2.76i·9-s + (0.336 − 0.336i)11-s + (0.318 + 0.318i)13-s + 1.33i·15-s + 1.38i·17-s + (0.787 − 0.787i)19-s + (1.83 − 0.638i)21-s − 0.469·23-s + 0.525i·25-s + (2.41 + 2.41i)27-s + (0.121 − 0.121i)29-s + 0.652·31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((−0.612−0.790i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=((−0.612−0.790i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: −0.612−0.790i-0.612 - 0.790i
Analytic conductor: 26.432826.4328
Root analytic conductor: 5.141285.14128
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ448(335,⋅)\chi_{448} (335, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 448, ( :3/2), −0.612−0.790i)(2,\ 448,\ (\ :3/2),\ -0.612 - 0.790i)

Particular Values

L(2)L(2) ≈\approx 0.72340194240.7234019424
L(12)L(\frac12) ≈\approx 0.72340194240.7234019424
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+(16.6+8.05i)T 1 + (16.6 + 8.05i)T
good3 1+(7.12−7.12i)T−27iT2 1 + (7.12 - 7.12i)T - 27iT^{2}
5 1+(−5.44+5.44i)T−125iT2 1 + (-5.44 + 5.44i)T - 125iT^{2}
11 1+(−12.2+12.2i)T−1.33e3iT2 1 + (-12.2 + 12.2i)T - 1.33e3iT^{2}
13 1+(−14.9−14.9i)T+2.19e3iT2 1 + (-14.9 - 14.9i)T + 2.19e3iT^{2}
17 1−97.2iT−4.91e3T2 1 - 97.2iT - 4.91e3T^{2}
19 1+(−65.2+65.2i)T−6.85e3iT2 1 + (-65.2 + 65.2i)T - 6.85e3iT^{2}
23 1+51.7T+1.21e4T2 1 + 51.7T + 1.21e4T^{2}
29 1+(−19.0+19.0i)T−2.43e4iT2 1 + (-19.0 + 19.0i)T - 2.43e4iT^{2}
31 1−112.T+2.97e4T2 1 - 112.T + 2.97e4T^{2}
37 1+(292.+292.i)T+5.06e4iT2 1 + (292. + 292. i)T + 5.06e4iT^{2}
41 1−145.T+6.89e4T2 1 - 145.T + 6.89e4T^{2}
43 1+(180.−180.i)T−7.95e4iT2 1 + (180. - 180. i)T - 7.95e4iT^{2}
47 1+325.T+1.03e5T2 1 + 325.T + 1.03e5T^{2}
53 1+(−19.7−19.7i)T+1.48e5iT2 1 + (-19.7 - 19.7i)T + 1.48e5iT^{2}
59 1+(−51.1−51.1i)T+2.05e5iT2 1 + (-51.1 - 51.1i)T + 2.05e5iT^{2}
61 1+(−286.−286.i)T+2.26e5iT2 1 + (-286. - 286. i)T + 2.26e5iT^{2}
67 1+(−529.−529.i)T+3.00e5iT2 1 + (-529. - 529. i)T + 3.00e5iT^{2}
71 1+246.T+3.57e5T2 1 + 246.T + 3.57e5T^{2}
73 1+798.T+3.89e5T2 1 + 798.T + 3.89e5T^{2}
79 1−629.iT−4.93e5T2 1 - 629. iT - 4.93e5T^{2}
83 1+(170.−170.i)T−5.71e5iT2 1 + (170. - 170. i)T - 5.71e5iT^{2}
89 1−113.T+7.04e5T2 1 - 113.T + 7.04e5T^{2}
97 1−1.41e3iT−9.12e5T2 1 - 1.41e3iT - 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

−10.89127745681539047375876176829, −10.11188692772097539066670059378, −9.500881333405775134728927196328, −8.720742016761683834710391541340, −6.85896485958697704121138871607, −6.06814783566427568584995632743, −5.36559205481661042697520023892, −4.23531900685831581931361338173, −3.47307255290809938020883620801, −1.01227218767807607058096187438, 0.34856389817251497167561884207, 1.72403387288010833285413668987, 2.96729031132632836308463206374, 4.99981223093447614656082634807, 5.88187638688759557265329199786, 6.57154079237715100650302773204, 7.18604099843084416213902652012, 8.294240438703296623326382130840, 9.791125458733655099280875369264, 10.39036482886498167926141701477

Graph of the ZZ-function along the critical line