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Properties

Label 2-384-128.101-c1-0-22
Degree 22
Conductor 384384
Sign 0.631+0.775i0.631 + 0.775i
Analytic cond. 3.066253.06625
Root an. cond. 1.751071.75107
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  + (0.827 − 1.14i)2-s + (0.881 − 0.471i)3-s + (−0.632 − 1.89i)4-s + (2.75 + 3.35i)5-s + (0.188 − 1.40i)6-s + (−1.44 + 0.963i)7-s + (−2.69 − 0.844i)8-s + (0.555 − 0.831i)9-s + (6.12 − 0.384i)10-s + (4.36 − 1.32i)11-s + (−1.45 − 1.37i)12-s + (−0.620 − 0.509i)13-s + (−0.0871 + 2.45i)14-s + (4.01 + 1.66i)15-s + (−3.20 + 2.39i)16-s + (2.30 − 0.956i)17-s + ⋯
L(s)  = 1  + (0.584 − 0.811i)2-s + (0.509 − 0.272i)3-s + (−0.316 − 0.948i)4-s + (1.23 + 1.50i)5-s + (0.0769 − 0.572i)6-s + (−0.544 + 0.364i)7-s + (−0.954 − 0.298i)8-s + (0.185 − 0.277i)9-s + (1.93 − 0.121i)10-s + (1.31 − 0.399i)11-s + (−0.419 − 0.397i)12-s + (−0.172 − 0.141i)13-s + (−0.0233 + 0.654i)14-s + (1.03 + 0.428i)15-s + (−0.800 + 0.599i)16-s + (0.560 − 0.232i)17-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.631+0.775i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((0.631+0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.631+0.775i0.631 + 0.775i
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ384(229,⋅)\chi_{384} (229, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), 0.631+0.775i)(2,\ 384,\ (\ :1/2),\ 0.631 + 0.775i)

Particular Values

L(1)L(1) ≈\approx 2.13190−1.01292i2.13190 - 1.01292i
L(12)L(\frac12) ≈\approx 2.13190−1.01292i2.13190 - 1.01292i
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+(−0.827+1.14i)T 1 + (-0.827 + 1.14i)T
3 1+(−0.881+0.471i)T 1 + (-0.881 + 0.471i)T
good5 1+(−2.75−3.35i)T+(−0.975+4.90i)T2 1 + (-2.75 - 3.35i)T + (-0.975 + 4.90i)T^{2}
7 1+(1.44−0.963i)T+(2.67−6.46i)T2 1 + (1.44 - 0.963i)T + (2.67 - 6.46i)T^{2}
11 1+(−4.36+1.32i)T+(9.14−6.11i)T2 1 + (-4.36 + 1.32i)T + (9.14 - 6.11i)T^{2}
13 1+(0.620+0.509i)T+(2.53+12.7i)T2 1 + (0.620 + 0.509i)T + (2.53 + 12.7i)T^{2}
17 1+(−2.30+0.956i)T+(12.0−12.0i)T2 1 + (-2.30 + 0.956i)T + (12.0 - 12.0i)T^{2}
19 1+(0.650+6.60i)T+(−18.6+3.70i)T2 1 + (0.650 + 6.60i)T + (-18.6 + 3.70i)T^{2}
23 1+(8.19−1.63i)T+(21.2−8.80i)T2 1 + (8.19 - 1.63i)T + (21.2 - 8.80i)T^{2}
29 1+(1.28−4.23i)T+(−24.1−16.1i)T2 1 + (1.28 - 4.23i)T + (-24.1 - 16.1i)T^{2}
31 1+(−0.608−0.608i)T+31iT2 1 + (-0.608 - 0.608i)T + 31iT^{2}
37 1+(5.62+0.553i)T+(36.2+7.21i)T2 1 + (5.62 + 0.553i)T + (36.2 + 7.21i)T^{2}
41 1+(0.680+3.42i)T+(−37.8+15.6i)T2 1 + (0.680 + 3.42i)T + (-37.8 + 15.6i)T^{2}
43 1+(−1.97−1.05i)T+(23.8+35.7i)T2 1 + (-1.97 - 1.05i)T + (23.8 + 35.7i)T^{2}
47 1+(1.97+4.77i)T+(−33.2+33.2i)T2 1 + (1.97 + 4.77i)T + (-33.2 + 33.2i)T^{2}
53 1+(3.50+11.5i)T+(−44.0+29.4i)T2 1 + (3.50 + 11.5i)T + (-44.0 + 29.4i)T^{2}
59 1+(0.265−0.218i)T+(11.5−57.8i)T2 1 + (0.265 - 0.218i)T + (11.5 - 57.8i)T^{2}
61 1+(−5.70−10.6i)T+(−33.8+50.7i)T2 1 + (-5.70 - 10.6i)T + (-33.8 + 50.7i)T^{2}
67 1+(−2.13−3.99i)T+(−37.2+55.7i)T2 1 + (-2.13 - 3.99i)T + (-37.2 + 55.7i)T^{2}
71 1+(1.73+2.59i)T+(−27.1+65.5i)T2 1 + (1.73 + 2.59i)T + (-27.1 + 65.5i)T^{2}
73 1+(3.87+2.58i)T+(27.9+67.4i)T2 1 + (3.87 + 2.58i)T + (27.9 + 67.4i)T^{2}
79 1+(1.12−2.72i)T+(−55.8−55.8i)T2 1 + (1.12 - 2.72i)T + (-55.8 - 55.8i)T^{2}
83 1+(3.76−0.370i)T+(81.4−16.1i)T2 1 + (3.76 - 0.370i)T + (81.4 - 16.1i)T^{2}
89 1+(−8.81−1.75i)T+(82.2+34.0i)T2 1 + (-8.81 - 1.75i)T + (82.2 + 34.0i)T^{2}
97 1+(−0.442−0.442i)T+97iT2 1 + (-0.442 - 0.442i)T + 97iT^{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

−11.25422495181704319376680497822, −10.25992566173305260329383173944, −9.614645777980867990623617005355, −8.908857728976771633082533844814, −7.00878672754113519024063429281, −6.38900732553524143584335214147, −5.51255119311623678734486702087, −3.67179583447648556678122991788, −2.86358009227586454001273466622, −1.85209650398546405236607924791, 1.82760940291487278531100829488, 3.78834890896637205637160571599, 4.53258547843263906922569121397, 5.78856726868836919874296266172, 6.38258498626172312885922292287, 7.87308436713083914146483051445, 8.644881762761940981761216626260, 9.612263467989309009561866324283, 9.952229940815430873788554411728, 12.10514503037445088491348565762

Graph of the ZZ-function along the critical line