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Properties

Label 2-448-112.27-c3-0-19
Degree 22
Conductor 448448
Sign 0.999+0.0395i0.999 + 0.0395i
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  + (−6.75 + 6.75i)3-s + (−10.7 + 10.7i)5-s + (−7.90 + 16.7i)7-s − 64.1i·9-s + (−5.28 + 5.28i)11-s + (−50.4 − 50.4i)13-s − 145. i·15-s − 17.6i·17-s + (−63.5 + 63.5i)19-s + (−59.6 − 166. i)21-s − 7.51·23-s − 105. i·25-s + (250. + 250. i)27-s + (−153. + 153. i)29-s − 60.8·31-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)3-s + (−0.960 + 0.960i)5-s + (−0.427 + 0.904i)7-s − 2.37i·9-s + (−0.144 + 0.144i)11-s + (−1.07 − 1.07i)13-s − 2.49i·15-s − 0.252i·17-s + (−0.767 + 0.767i)19-s + (−0.619 − 1.72i)21-s − 0.0681·23-s − 0.846i·25-s + (1.78 + 1.78i)27-s + (−0.985 + 0.985i)29-s − 0.352·31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((0.999+0.0395i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0395i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=((0.999+0.0395i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 0.999+0.0395i0.999 + 0.0395i
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), 0.999+0.0395i)(2,\ 448,\ (\ :3/2),\ 0.999 + 0.0395i)

Particular Values

L(2)L(2) ≈\approx 0.041378434850.04137843485
L(12)L(\frac12) ≈\approx 0.041378434850.04137843485
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+(7.90−16.7i)T 1 + (7.90 - 16.7i)T
good3 1+(6.75−6.75i)T−27iT2 1 + (6.75 - 6.75i)T - 27iT^{2}
5 1+(10.7−10.7i)T−125iT2 1 + (10.7 - 10.7i)T - 125iT^{2}
11 1+(5.28−5.28i)T−1.33e3iT2 1 + (5.28 - 5.28i)T - 1.33e3iT^{2}
13 1+(50.4+50.4i)T+2.19e3iT2 1 + (50.4 + 50.4i)T + 2.19e3iT^{2}
17 1+17.6iT−4.91e3T2 1 + 17.6iT - 4.91e3T^{2}
19 1+(63.5−63.5i)T−6.85e3iT2 1 + (63.5 - 63.5i)T - 6.85e3iT^{2}
23 1+7.51T+1.21e4T2 1 + 7.51T + 1.21e4T^{2}
29 1+(153.−153.i)T−2.43e4iT2 1 + (153. - 153. i)T - 2.43e4iT^{2}
31 1+60.8T+2.97e4T2 1 + 60.8T + 2.97e4T^{2}
37 1+(−129.−129.i)T+5.06e4iT2 1 + (-129. - 129. i)T + 5.06e4iT^{2}
41 1−48.6T+6.89e4T2 1 - 48.6T + 6.89e4T^{2}
43 1+(123.−123.i)T−7.95e4iT2 1 + (123. - 123. i)T - 7.95e4iT^{2}
47 1+254.T+1.03e5T2 1 + 254.T + 1.03e5T^{2}
53 1+(−498.−498.i)T+1.48e5iT2 1 + (-498. - 498. i)T + 1.48e5iT^{2}
59 1+(−122.−122.i)T+2.05e5iT2 1 + (-122. - 122. i)T + 2.05e5iT^{2}
61 1+(228.+228.i)T+2.26e5iT2 1 + (228. + 228. i)T + 2.26e5iT^{2}
67 1+(360.+360.i)T+3.00e5iT2 1 + (360. + 360. i)T + 3.00e5iT^{2}
71 1+605.T+3.57e5T2 1 + 605.T + 3.57e5T^{2}
73 1−913.T+3.89e5T2 1 - 913.T + 3.89e5T^{2}
79 1−885.iT−4.93e5T2 1 - 885. iT - 4.93e5T^{2}
83 1+(108.−108.i)T−5.71e5iT2 1 + (108. - 108. i)T - 5.71e5iT^{2}
89 1+269.T+7.04e5T2 1 + 269.T + 7.04e5T^{2}
97 1−1.45e3iT−9.12e5T2 1 - 1.45e3iT - 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

−10.66694686704496074244642312148, −10.07235899869138423419461088412, −9.197864232170518225608309127616, −7.84595461721996450741293141735, −6.74174274762740360651901113830, −5.77785185973644303040770963721, −4.98562495244843630009036247703, −3.82465526925826568247347325467, −2.90250903419116486877727288460, −0.03051687873364960910609745936, 0.57103944575874449199509264025, 1.98492641401225766786043181846, 4.15061823331898492087020137123, 4.92062912731103606876707471231, 6.12877036603204872036762556210, 7.11265519225913386185398051432, 7.55389789446307264360801178847, 8.638881895160060332357096722386, 9.948973927322909245990885981097, 11.12816303560503375323555774005

Graph of the ZZ-function along the critical line