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Properties

Label 2-384-128.101-c1-0-17
Degree 22
Conductor 384384
Sign 0.876−0.480i0.876 - 0.480i
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  + (1.38 + 0.283i)2-s + (−0.881 + 0.471i)3-s + (1.83 + 0.785i)4-s + (−0.311 − 0.379i)5-s + (−1.35 + 0.403i)6-s + (2.05 − 1.37i)7-s + (2.32 + 1.60i)8-s + (0.555 − 0.831i)9-s + (−0.324 − 0.614i)10-s + (0.708 − 0.215i)11-s + (−1.99 + 0.174i)12-s + (1.87 + 1.54i)13-s + (3.24 − 1.32i)14-s + (0.453 + 0.187i)15-s + (2.76 + 2.88i)16-s + (0.338 − 0.140i)17-s + ⋯
L(s)  = 1  + (0.979 + 0.200i)2-s + (−0.509 + 0.272i)3-s + (0.919 + 0.392i)4-s + (−0.139 − 0.169i)5-s + (−0.553 + 0.164i)6-s + (0.777 − 0.519i)7-s + (0.822 + 0.569i)8-s + (0.185 − 0.277i)9-s + (−0.102 − 0.194i)10-s + (0.213 − 0.0648i)11-s + (−0.575 + 0.0502i)12-s + (0.521 + 0.427i)13-s + (0.866 − 0.353i)14-s + (0.117 + 0.0485i)15-s + (0.691 + 0.722i)16-s + (0.0822 − 0.0340i)17-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.876−0.480i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((0.876−0.480i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.876−0.480i0.876 - 0.480i
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), 0.876−0.480i)(2,\ 384,\ (\ :1/2),\ 0.876 - 0.480i)

Particular Values

L(1)L(1) ≈\approx 2.18586+0.559941i2.18586 + 0.559941i
L(12)L(\frac12) ≈\approx 2.18586+0.559941i2.18586 + 0.559941i
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+(−1.38−0.283i)T 1 + (-1.38 - 0.283i)T
3 1+(0.881−0.471i)T 1 + (0.881 - 0.471i)T
good5 1+(0.311+0.379i)T+(−0.975+4.90i)T2 1 + (0.311 + 0.379i)T + (-0.975 + 4.90i)T^{2}
7 1+(−2.05+1.37i)T+(2.67−6.46i)T2 1 + (-2.05 + 1.37i)T + (2.67 - 6.46i)T^{2}
11 1+(−0.708+0.215i)T+(9.14−6.11i)T2 1 + (-0.708 + 0.215i)T + (9.14 - 6.11i)T^{2}
13 1+(−1.87−1.54i)T+(2.53+12.7i)T2 1 + (-1.87 - 1.54i)T + (2.53 + 12.7i)T^{2}
17 1+(−0.338+0.140i)T+(12.0−12.0i)T2 1 + (-0.338 + 0.140i)T + (12.0 - 12.0i)T^{2}
19 1+(−0.634−6.44i)T+(−18.6+3.70i)T2 1 + (-0.634 - 6.44i)T + (-18.6 + 3.70i)T^{2}
23 1+(7.28−1.44i)T+(21.2−8.80i)T2 1 + (7.28 - 1.44i)T + (21.2 - 8.80i)T^{2}
29 1+(−0.561+1.84i)T+(−24.1−16.1i)T2 1 + (-0.561 + 1.84i)T + (-24.1 - 16.1i)T^{2}
31 1+(5.57+5.57i)T+31iT2 1 + (5.57 + 5.57i)T + 31iT^{2}
37 1+(−6.92−0.681i)T+(36.2+7.21i)T2 1 + (-6.92 - 0.681i)T + (36.2 + 7.21i)T^{2}
41 1+(0.989+4.97i)T+(−37.8+15.6i)T2 1 + (0.989 + 4.97i)T + (-37.8 + 15.6i)T^{2}
43 1+(5.69+3.04i)T+(23.8+35.7i)T2 1 + (5.69 + 3.04i)T + (23.8 + 35.7i)T^{2}
47 1+(1.01+2.44i)T+(−33.2+33.2i)T2 1 + (1.01 + 2.44i)T + (-33.2 + 33.2i)T^{2}
53 1+(2.92+9.62i)T+(−44.0+29.4i)T2 1 + (2.92 + 9.62i)T + (-44.0 + 29.4i)T^{2}
59 1+(8.85−7.26i)T+(11.5−57.8i)T2 1 + (8.85 - 7.26i)T + (11.5 - 57.8i)T^{2}
61 1+(−0.238−0.445i)T+(−33.8+50.7i)T2 1 + (-0.238 - 0.445i)T + (-33.8 + 50.7i)T^{2}
67 1+(−1.29−2.41i)T+(−37.2+55.7i)T2 1 + (-1.29 - 2.41i)T + (-37.2 + 55.7i)T^{2}
71 1+(−0.719−1.07i)T+(−27.1+65.5i)T2 1 + (-0.719 - 1.07i)T + (-27.1 + 65.5i)T^{2}
73 1+(0.675+0.451i)T+(27.9+67.4i)T2 1 + (0.675 + 0.451i)T + (27.9 + 67.4i)T^{2}
79 1+(−3.41+8.25i)T+(−55.8−55.8i)T2 1 + (-3.41 + 8.25i)T + (-55.8 - 55.8i)T^{2}
83 1+(11.2−1.11i)T+(81.4−16.1i)T2 1 + (11.2 - 1.11i)T + (81.4 - 16.1i)T^{2}
89 1+(9.94+1.97i)T+(82.2+34.0i)T2 1 + (9.94 + 1.97i)T + (82.2 + 34.0i)T^{2}
97 1+(−1.02−1.02i)T+97iT2 1 + (-1.02 - 1.02i)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

−11.64378136740116937443839286396, −10.75877595514292558268105182763, −9.885987810078444797782193560804, −8.302989762358513271469142061791, −7.59467055968497960718331787413, −6.34003483580751085176651148794, −5.59364424426620199676162935857, −4.36731414447397830681258373694, −3.77687349066173742865966072683, −1.77710229957004080504139516424, 1.60411158065659281031962977796, 3.05127339966517342619182212923, 4.47712683045955537079779590812, 5.35538030299224001002000704031, 6.27853083913904462674373850411, 7.26137764927665780460418201697, 8.285875591158980316965877695559, 9.646090024940872781561859601972, 10.97597744798518270507786180914, 11.22290589663022689618589628469

Graph of the ZZ-function along the critical line