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Properties

Label 2-448-112.27-c3-0-12
Degree 22
Conductor 448448
Sign 0.0269−0.999i0.0269 - 0.999i
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  + (4.79 − 4.79i)3-s + (2.04 − 2.04i)5-s + (−13.3 + 12.8i)7-s − 19.0i·9-s + (−34.5 + 34.5i)11-s + (−7.32 − 7.32i)13-s − 19.6i·15-s + 133. i·17-s + (13.6 − 13.6i)19-s + (−2.69 + 125. i)21-s − 129.·23-s + 116. i·25-s + (38.2 + 38.2i)27-s + (85.7 − 85.7i)29-s − 239.·31-s + ⋯
L(s)  = 1  + (0.923 − 0.923i)3-s + (0.183 − 0.183i)5-s + (−0.722 + 0.691i)7-s − 0.704i·9-s + (−0.947 + 0.947i)11-s + (−0.156 − 0.156i)13-s − 0.337i·15-s + 1.90i·17-s + (0.165 − 0.165i)19-s + (−0.0280 + 1.30i)21-s − 1.17·23-s + 0.933i·25-s + (0.272 + 0.272i)27-s + (0.549 − 0.549i)29-s − 1.38·31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((0.0269−0.999i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0269 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=((0.0269−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0269 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 0.0269−0.999i0.0269 - 0.999i
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), 0.0269−0.999i)(2,\ 448,\ (\ :3/2),\ 0.0269 - 0.999i)

Particular Values

L(2)L(2) ≈\approx 1.3126400441.312640044
L(12)L(\frac12) ≈\approx 1.3126400441.312640044
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+(13.3−12.8i)T 1 + (13.3 - 12.8i)T
good3 1+(−4.79+4.79i)T−27iT2 1 + (-4.79 + 4.79i)T - 27iT^{2}
5 1+(−2.04+2.04i)T−125iT2 1 + (-2.04 + 2.04i)T - 125iT^{2}
11 1+(34.5−34.5i)T−1.33e3iT2 1 + (34.5 - 34.5i)T - 1.33e3iT^{2}
13 1+(7.32+7.32i)T+2.19e3iT2 1 + (7.32 + 7.32i)T + 2.19e3iT^{2}
17 1−133.iT−4.91e3T2 1 - 133. iT - 4.91e3T^{2}
19 1+(−13.6+13.6i)T−6.85e3iT2 1 + (-13.6 + 13.6i)T - 6.85e3iT^{2}
23 1+129.T+1.21e4T2 1 + 129.T + 1.21e4T^{2}
29 1+(−85.7+85.7i)T−2.43e4iT2 1 + (-85.7 + 85.7i)T - 2.43e4iT^{2}
31 1+239.T+2.97e4T2 1 + 239.T + 2.97e4T^{2}
37 1+(50.7+50.7i)T+5.06e4iT2 1 + (50.7 + 50.7i)T + 5.06e4iT^{2}
41 1−118.T+6.89e4T2 1 - 118.T + 6.89e4T^{2}
43 1+(−75.4+75.4i)T−7.95e4iT2 1 + (-75.4 + 75.4i)T - 7.95e4iT^{2}
47 1+169.T+1.03e5T2 1 + 169.T + 1.03e5T^{2}
53 1+(−482.−482.i)T+1.48e5iT2 1 + (-482. - 482. i)T + 1.48e5iT^{2}
59 1+(285.+285.i)T+2.05e5iT2 1 + (285. + 285. i)T + 2.05e5iT^{2}
61 1+(313.+313.i)T+2.26e5iT2 1 + (313. + 313. i)T + 2.26e5iT^{2}
67 1+(−58.2−58.2i)T+3.00e5iT2 1 + (-58.2 - 58.2i)T + 3.00e5iT^{2}
71 1−374.T+3.57e5T2 1 - 374.T + 3.57e5T^{2}
73 1−610.T+3.89e5T2 1 - 610.T + 3.89e5T^{2}
79 1+460.iT−4.93e5T2 1 + 460. iT - 4.93e5T^{2}
83 1+(875.−875.i)T−5.71e5iT2 1 + (875. - 875. i)T - 5.71e5iT^{2}
89 1+633.T+7.04e5T2 1 + 633.T + 7.04e5T^{2}
97 1−946.iT−9.12e5T2 1 - 946. iT - 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

−10.75693299054901095285444095094, −9.855197971114826224853166091399, −8.972950148497842853764088850987, −8.092307078938678598665513165559, −7.43096483506904223638606777507, −6.31845339661501271025994028451, −5.35275721410005966676253186220, −3.76879594750194334726074498541, −2.50364474796370093104980852190, −1.75352328209435230274401789300, 0.34387728702699672621786074023, 2.62294181190414352673073012814, 3.33903203546686801120450918206, 4.39913309126112751627191001369, 5.57982172293140895359950382624, 6.84583198064851943439785072054, 7.84393114925619760861459518231, 8.805027103640597731720723790986, 9.685127180234239825726943108466, 10.19362585649302882391907748313

Graph of the ZZ-function along the critical line