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Properties

Label 2-448-112.19-c1-0-13
Degree 22
Conductor 448448
Sign −0.927−0.373i-0.927 - 0.373i
Analytic cond. 3.577293.57729
Root an. cond. 1.891371.89137
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.665 − 2.48i)3-s + (−3.12 + 0.837i)5-s + (−1.56 − 2.13i)7-s + (−3.13 − 1.80i)9-s + (−0.376 + 1.40i)11-s + (−3.11 + 3.11i)13-s + 8.31i·15-s + (2.02 − 1.16i)17-s + (−4.40 + 1.18i)19-s + (−6.34 + 2.45i)21-s + (1.15 − 1.99i)23-s + (4.72 − 2.73i)25-s + (−1.12 + 1.12i)27-s + (−1.55 − 1.55i)29-s + (−3.88 − 6.73i)31-s + ⋯
L(s)  = 1  + (0.384 − 1.43i)3-s + (−1.39 + 0.374i)5-s + (−0.590 − 0.807i)7-s + (−1.04 − 0.602i)9-s + (−0.113 + 0.423i)11-s + (−0.863 + 0.863i)13-s + 2.14i·15-s + (0.490 − 0.283i)17-s + (−1.01 + 0.270i)19-s + (−1.38 + 0.536i)21-s + (0.240 − 0.416i)23-s + (0.945 − 0.546i)25-s + (−0.216 + 0.216i)27-s + (−0.288 − 0.288i)29-s + (−0.698 − 1.20i)31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((−0.927−0.373i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+1/2) L(s)=((−0.927−0.373i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: −0.927−0.373i-0.927 - 0.373i
Analytic conductor: 3.577293.57729
Root analytic conductor: 1.891371.89137
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ448(47,⋅)\chi_{448} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 448, ( :1/2), −0.927−0.373i)(2,\ 448,\ (\ :1/2),\ -0.927 - 0.373i)

Particular Values

L(1)L(1) ≈\approx 0.0803259+0.414527i0.0803259 + 0.414527i
L(12)L(\frac12) ≈\approx 0.0803259+0.414527i0.0803259 + 0.414527i
L(32)L(\frac{3}{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+(1.56+2.13i)T 1 + (1.56 + 2.13i)T
good3 1+(−0.665+2.48i)T+(−2.59−1.5i)T2 1 + (-0.665 + 2.48i)T + (-2.59 - 1.5i)T^{2}
5 1+(3.12−0.837i)T+(4.33−2.5i)T2 1 + (3.12 - 0.837i)T + (4.33 - 2.5i)T^{2}
11 1+(0.376−1.40i)T+(−9.52−5.5i)T2 1 + (0.376 - 1.40i)T + (-9.52 - 5.5i)T^{2}
13 1+(3.11−3.11i)T−13iT2 1 + (3.11 - 3.11i)T - 13iT^{2}
17 1+(−2.02+1.16i)T+(8.5−14.7i)T2 1 + (-2.02 + 1.16i)T + (8.5 - 14.7i)T^{2}
19 1+(4.40−1.18i)T+(16.4−9.5i)T2 1 + (4.40 - 1.18i)T + (16.4 - 9.5i)T^{2}
23 1+(−1.15+1.99i)T+(−11.5−19.9i)T2 1 + (-1.15 + 1.99i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.55+1.55i)T+29iT2 1 + (1.55 + 1.55i)T + 29iT^{2}
31 1+(3.88+6.73i)T+(−15.5+26.8i)T2 1 + (3.88 + 6.73i)T + (-15.5 + 26.8i)T^{2}
37 1+(−0.272−1.01i)T+(−32.0+18.5i)T2 1 + (-0.272 - 1.01i)T + (-32.0 + 18.5i)T^{2}
41 1+2.77T+41T2 1 + 2.77T + 41T^{2}
43 1+(7.12+7.12i)T+43iT2 1 + (7.12 + 7.12i)T + 43iT^{2}
47 1+(1.42−2.46i)T+(−23.5−40.7i)T2 1 + (1.42 - 2.46i)T + (-23.5 - 40.7i)T^{2}
53 1+(−11.0−2.97i)T+(45.8+26.5i)T2 1 + (-11.0 - 2.97i)T + (45.8 + 26.5i)T^{2}
59 1+(3.77+1.01i)T+(51.0+29.5i)T2 1 + (3.77 + 1.01i)T + (51.0 + 29.5i)T^{2}
61 1+(3.72+13.9i)T+(−52.8+30.5i)T2 1 + (3.72 + 13.9i)T + (-52.8 + 30.5i)T^{2}
67 1+(2.59+0.695i)T+(58.0+33.5i)T2 1 + (2.59 + 0.695i)T + (58.0 + 33.5i)T^{2}
71 1−7.48T+71T2 1 - 7.48T + 71T^{2}
73 1+(5.65+9.78i)T+(−36.5+63.2i)T2 1 + (5.65 + 9.78i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.706−0.408i)T+(39.5+68.4i)T2 1 + (-0.706 - 0.408i)T + (39.5 + 68.4i)T^{2}
83 1+(2.65+2.65i)T+83iT2 1 + (2.65 + 2.65i)T + 83iT^{2}
89 1+(2.40−4.17i)T+(−44.5−77.0i)T2 1 + (2.40 - 4.17i)T + (-44.5 - 77.0i)T^{2}
97 1−5.86iT−97T2 1 - 5.86iT - 97T^{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

−10.75974879451346039553347993320, −9.658449082759752137631413670654, −8.415609813707422741717543105714, −7.52432068399576980876082865641, −7.18786666929448425897281250116, −6.37811109805106989261475756159, −4.49609084119022581310311235328, −3.46835486507065191959339141094, −2.12308148251220743799365354808, −0.24006533451651154365243938985, 2.96715074306237975754203021045, 3.67797766220875716717560347850, 4.74459390888821365599454511838, 5.58347944070559178104227365306, 7.17639119538536659417825159577, 8.370262961468286556414405267204, 8.781835807267608284618117744447, 9.836259236225720693393966594179, 10.57013531816546940079818015741, 11.51563421222398275964935475877

Graph of the ZZ-function along the critical line