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Properties

Label 2-448-112.27-c3-0-20
Degree 22
Conductor 448448
Sign 0.781−0.623i0.781 - 0.623i
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  + (3.57 − 3.57i)3-s + (2.98 − 2.98i)5-s + (−17.0 + 7.14i)7-s + 1.45i·9-s + (−10.4 + 10.4i)11-s + (53.8 + 53.8i)13-s − 21.3i·15-s − 29.4i·17-s + (−68.1 + 68.1i)19-s + (−35.5 + 86.6i)21-s + 108.·23-s + 107. i·25-s + (101. + 101. i)27-s + (−36.5 + 36.5i)29-s + 287.·31-s + ⋯
L(s)  = 1  + (0.687 − 0.687i)3-s + (0.267 − 0.267i)5-s + (−0.922 + 0.385i)7-s + 0.0538i·9-s + (−0.287 + 0.287i)11-s + (1.14 + 1.14i)13-s − 0.367i·15-s − 0.420i·17-s + (−0.822 + 0.822i)19-s + (−0.369 + 0.899i)21-s + 0.982·23-s + 0.857i·25-s + (0.724 + 0.724i)27-s + (−0.233 + 0.233i)29-s + 1.66·31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((0.781−0.623i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=((0.781−0.623i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 0.781−0.623i0.781 - 0.623i
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), 0.781−0.623i)(2,\ 448,\ (\ :3/2),\ 0.781 - 0.623i)

Particular Values

L(2)L(2) ≈\approx 2.1257348302.125734830
L(12)L(\frac12) ≈\approx 2.1257348302.125734830
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+(17.0−7.14i)T 1 + (17.0 - 7.14i)T
good3 1+(−3.57+3.57i)T−27iT2 1 + (-3.57 + 3.57i)T - 27iT^{2}
5 1+(−2.98+2.98i)T−125iT2 1 + (-2.98 + 2.98i)T - 125iT^{2}
11 1+(10.4−10.4i)T−1.33e3iT2 1 + (10.4 - 10.4i)T - 1.33e3iT^{2}
13 1+(−53.8−53.8i)T+2.19e3iT2 1 + (-53.8 - 53.8i)T + 2.19e3iT^{2}
17 1+29.4iT−4.91e3T2 1 + 29.4iT - 4.91e3T^{2}
19 1+(68.1−68.1i)T−6.85e3iT2 1 + (68.1 - 68.1i)T - 6.85e3iT^{2}
23 1−108.T+1.21e4T2 1 - 108.T + 1.21e4T^{2}
29 1+(36.5−36.5i)T−2.43e4iT2 1 + (36.5 - 36.5i)T - 2.43e4iT^{2}
31 1−287.T+2.97e4T2 1 - 287.T + 2.97e4T^{2}
37 1+(−38.5−38.5i)T+5.06e4iT2 1 + (-38.5 - 38.5i)T + 5.06e4iT^{2}
41 1+315.T+6.89e4T2 1 + 315.T + 6.89e4T^{2}
43 1+(−182.+182.i)T−7.95e4iT2 1 + (-182. + 182. i)T - 7.95e4iT^{2}
47 1+323.T+1.03e5T2 1 + 323.T + 1.03e5T^{2}
53 1+(8.29+8.29i)T+1.48e5iT2 1 + (8.29 + 8.29i)T + 1.48e5iT^{2}
59 1+(246.+246.i)T+2.05e5iT2 1 + (246. + 246. i)T + 2.05e5iT^{2}
61 1+(−289.−289.i)T+2.26e5iT2 1 + (-289. - 289. i)T + 2.26e5iT^{2}
67 1+(−98.9−98.9i)T+3.00e5iT2 1 + (-98.9 - 98.9i)T + 3.00e5iT^{2}
71 1−1.13e3T+3.57e5T2 1 - 1.13e3T + 3.57e5T^{2}
73 1+653.T+3.89e5T2 1 + 653.T + 3.89e5T^{2}
79 1−486.iT−4.93e5T2 1 - 486. iT - 4.93e5T^{2}
83 1+(141.−141.i)T−5.71e5iT2 1 + (141. - 141. i)T - 5.71e5iT^{2}
89 1−497.T+7.04e5T2 1 - 497.T + 7.04e5T^{2}
97 1−1.55e3iT−9.12e5T2 1 - 1.55e3iT - 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

−10.74197123480206663558755826840, −9.670823537621374121776216011503, −8.869040935135564863334554027963, −8.206257684296474221979006545086, −6.98294978351606567266074300146, −6.33739918265368080884330564392, −5.07204778067798915629597999489, −3.65378164169492731276992909307, −2.48570065187673339042116768930, −1.39093068284234384183037432581, 0.66093501095708459165050692015, 2.77865050937312623964575092437, 3.42880971789184737438167811926, 4.53950460838578357897980774561, 6.03596301177167297319127422891, 6.67875008088306863128661055768, 8.150358340228916385479541664568, 8.771150391222301802350179626320, 9.808264674951821521486682486439, 10.39043438092803356682339045775

Graph of the ZZ-function along the critical line