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Properties

Label 2-384-128.101-c1-0-16
Degree 22
Conductor 384384
Sign 0.870−0.492i0.870 - 0.492i
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  + (−1.08 + 0.908i)2-s + (0.881 − 0.471i)3-s + (0.349 − 1.96i)4-s + (1.51 + 1.84i)5-s + (−0.527 + 1.31i)6-s + (1.69 − 1.13i)7-s + (1.41 + 2.45i)8-s + (0.555 − 0.831i)9-s + (−3.31 − 0.623i)10-s + (1.92 − 0.585i)11-s + (−0.620 − 1.90i)12-s + (−1.76 − 1.44i)13-s + (−0.809 + 2.77i)14-s + (2.20 + 0.912i)15-s + (−3.75 − 1.37i)16-s + (−1.20 + 0.497i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.509 − 0.272i)3-s + (0.174 − 0.984i)4-s + (0.676 + 0.824i)5-s + (−0.215 + 0.535i)6-s + (0.641 − 0.428i)7-s + (0.498 + 0.866i)8-s + (0.185 − 0.277i)9-s + (−1.04 − 0.197i)10-s + (0.581 − 0.176i)11-s + (−0.179 − 0.548i)12-s + (−0.488 − 0.401i)13-s + (−0.216 + 0.740i)14-s + (0.568 + 0.235i)15-s + (−0.939 − 0.343i)16-s + (−0.291 + 0.120i)17-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.870−0.492i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((0.870−0.492i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.870−0.492i0.870 - 0.492i
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), 0.870−0.492i)(2,\ 384,\ (\ :1/2),\ 0.870 - 0.492i)

Particular Values

L(1)L(1) ≈\approx 1.29679+0.341297i1.29679 + 0.341297i
L(12)L(\frac12) ≈\approx 1.29679+0.341297i1.29679 + 0.341297i
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+(1.08−0.908i)T 1 + (1.08 - 0.908i)T
3 1+(−0.881+0.471i)T 1 + (-0.881 + 0.471i)T
good5 1+(−1.51−1.84i)T+(−0.975+4.90i)T2 1 + (-1.51 - 1.84i)T + (-0.975 + 4.90i)T^{2}
7 1+(−1.69+1.13i)T+(2.67−6.46i)T2 1 + (-1.69 + 1.13i)T + (2.67 - 6.46i)T^{2}
11 1+(−1.92+0.585i)T+(9.14−6.11i)T2 1 + (-1.92 + 0.585i)T + (9.14 - 6.11i)T^{2}
13 1+(1.76+1.44i)T+(2.53+12.7i)T2 1 + (1.76 + 1.44i)T + (2.53 + 12.7i)T^{2}
17 1+(1.20−0.497i)T+(12.0−12.0i)T2 1 + (1.20 - 0.497i)T + (12.0 - 12.0i)T^{2}
19 1+(−0.0663−0.674i)T+(−18.6+3.70i)T2 1 + (-0.0663 - 0.674i)T + (-18.6 + 3.70i)T^{2}
23 1+(−4.83+0.962i)T+(21.2−8.80i)T2 1 + (-4.83 + 0.962i)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+(−3.94−3.94i)T+31iT2 1 + (-3.94 - 3.94i)T + 31iT^{2}
37 1+(−1.45−0.142i)T+(36.2+7.21i)T2 1 + (-1.45 - 0.142i)T + (36.2 + 7.21i)T^{2}
41 1+(1.24+6.25i)T+(−37.8+15.6i)T2 1 + (1.24 + 6.25i)T + (-37.8 + 15.6i)T^{2}
43 1+(3.63+1.94i)T+(23.8+35.7i)T2 1 + (3.63 + 1.94i)T + (23.8 + 35.7i)T^{2}
47 1+(−3.99−9.64i)T+(−33.2+33.2i)T2 1 + (-3.99 - 9.64i)T + (-33.2 + 33.2i)T^{2}
53 1+(−1.42−4.70i)T+(−44.0+29.4i)T2 1 + (-1.42 - 4.70i)T + (-44.0 + 29.4i)T^{2}
59 1+(6.73−5.52i)T+(11.5−57.8i)T2 1 + (6.73 - 5.52i)T + (11.5 - 57.8i)T^{2}
61 1+(3.92+7.33i)T+(−33.8+50.7i)T2 1 + (3.92 + 7.33i)T + (-33.8 + 50.7i)T^{2}
67 1+(−2.13−4.00i)T+(−37.2+55.7i)T2 1 + (-2.13 - 4.00i)T + (-37.2 + 55.7i)T^{2}
71 1+(8.27+12.3i)T+(−27.1+65.5i)T2 1 + (8.27 + 12.3i)T + (-27.1 + 65.5i)T^{2}
73 1+(6.01+4.02i)T+(27.9+67.4i)T2 1 + (6.01 + 4.02i)T + (27.9 + 67.4i)T^{2}
79 1+(−2.42+5.85i)T+(−55.8−55.8i)T2 1 + (-2.42 + 5.85i)T + (-55.8 - 55.8i)T^{2}
83 1+(−4.16+0.409i)T+(81.4−16.1i)T2 1 + (-4.16 + 0.409i)T + (81.4 - 16.1i)T^{2}
89 1+(10.0+2.00i)T+(82.2+34.0i)T2 1 + (10.0 + 2.00i)T + (82.2 + 34.0i)T^{2}
97 1+(9.11+9.11i)T+97iT2 1 + (9.11 + 9.11i)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

−10.92520572451154792807093349213, −10.47993662226547525162508363219, −9.427171034747286371743001050984, −8.643853726747585331095823774330, −7.59237242254582334927247543761, −6.89656447252691753048375116774, −6.00490625267053477440643553331, −4.69591440042243274562814112047, −2.85287051553552300000232685140, −1.46903793636015915168353248998, 1.48345669559854063054370597819, 2.58117435809163873297646277199, 4.16890328982567681333491604410, 5.16128707621665696781467200875, 6.75667935636142361170451516771, 7.940662310780939395228253269064, 8.795020550191700814754991516281, 9.376732415765089984485261301555, 10.05149091346186008132947851139, 11.32894834101758905661544787336

Graph of the ZZ-function along the critical line