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Properties

Label 2-384-128.101-c1-0-23
Degree 22
Conductor 384384
Sign 0.472+0.881i0.472 + 0.881i
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  + (1.26 − 0.640i)2-s + (−0.881 + 0.471i)3-s + (1.17 − 1.61i)4-s + (−0.276 − 0.336i)5-s + (−0.810 + 1.15i)6-s + (0.338 − 0.226i)7-s + (0.453 − 2.79i)8-s + (0.555 − 0.831i)9-s + (−0.563 − 0.247i)10-s + (5.34 − 1.62i)11-s + (−0.279 + 1.98i)12-s + (−0.687 − 0.563i)13-s + (0.282 − 0.502i)14-s + (0.402 + 0.166i)15-s + (−1.21 − 3.81i)16-s + (−2.42 + 1.00i)17-s + ⋯
L(s)  = 1  + (0.891 − 0.452i)2-s + (−0.509 + 0.272i)3-s + (0.589 − 0.807i)4-s + (−0.123 − 0.150i)5-s + (−0.330 + 0.473i)6-s + (0.128 − 0.0855i)7-s + (0.160 − 0.987i)8-s + (0.185 − 0.277i)9-s + (−0.178 − 0.0782i)10-s + (1.61 − 0.488i)11-s + (−0.0805 + 0.571i)12-s + (−0.190 − 0.156i)13-s + (0.0754 − 0.134i)14-s + (0.103 + 0.0430i)15-s + (−0.304 − 0.952i)16-s + (−0.588 + 0.243i)17-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.472+0.881i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((0.472+0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.472+0.881i0.472 + 0.881i
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), 0.472+0.881i)(2,\ 384,\ (\ :1/2),\ 0.472 + 0.881i)

Particular Values

L(1)L(1) ≈\approx 1.71768−1.02861i1.71768 - 1.02861i
L(12)L(\frac12) ≈\approx 1.71768−1.02861i1.71768 - 1.02861i
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+(−1.26+0.640i)T 1 + (-1.26 + 0.640i)T
3 1+(0.881−0.471i)T 1 + (0.881 - 0.471i)T
good5 1+(0.276+0.336i)T+(−0.975+4.90i)T2 1 + (0.276 + 0.336i)T + (-0.975 + 4.90i)T^{2}
7 1+(−0.338+0.226i)T+(2.67−6.46i)T2 1 + (-0.338 + 0.226i)T + (2.67 - 6.46i)T^{2}
11 1+(−5.34+1.62i)T+(9.14−6.11i)T2 1 + (-5.34 + 1.62i)T + (9.14 - 6.11i)T^{2}
13 1+(0.687+0.563i)T+(2.53+12.7i)T2 1 + (0.687 + 0.563i)T + (2.53 + 12.7i)T^{2}
17 1+(2.42−1.00i)T+(12.0−12.0i)T2 1 + (2.42 - 1.00i)T + (12.0 - 12.0i)T^{2}
19 1+(0.274+2.78i)T+(−18.6+3.70i)T2 1 + (0.274 + 2.78i)T + (-18.6 + 3.70i)T^{2}
23 1+(−2.38+0.473i)T+(21.2−8.80i)T2 1 + (-2.38 + 0.473i)T + (21.2 - 8.80i)T^{2}
29 1+(2.19−7.25i)T+(−24.1−16.1i)T2 1 + (2.19 - 7.25i)T + (-24.1 - 16.1i)T^{2}
31 1+(−6.54−6.54i)T+31iT2 1 + (-6.54 - 6.54i)T + 31iT^{2}
37 1+(8.61+0.848i)T+(36.2+7.21i)T2 1 + (8.61 + 0.848i)T + (36.2 + 7.21i)T^{2}
41 1+(−1.68−8.49i)T+(−37.8+15.6i)T2 1 + (-1.68 - 8.49i)T + (-37.8 + 15.6i)T^{2}
43 1+(3.39+1.81i)T+(23.8+35.7i)T2 1 + (3.39 + 1.81i)T + (23.8 + 35.7i)T^{2}
47 1+(−2.17−5.24i)T+(−33.2+33.2i)T2 1 + (-2.17 - 5.24i)T + (-33.2 + 33.2i)T^{2}
53 1+(−0.0205−0.0676i)T+(−44.0+29.4i)T2 1 + (-0.0205 - 0.0676i)T + (-44.0 + 29.4i)T^{2}
59 1+(2.24−1.83i)T+(11.5−57.8i)T2 1 + (2.24 - 1.83i)T + (11.5 - 57.8i)T^{2}
61 1+(2.98+5.57i)T+(−33.8+50.7i)T2 1 + (2.98 + 5.57i)T + (-33.8 + 50.7i)T^{2}
67 1+(3.71+6.94i)T+(−37.2+55.7i)T2 1 + (3.71 + 6.94i)T + (-37.2 + 55.7i)T^{2}
71 1+(−5.59−8.37i)T+(−27.1+65.5i)T2 1 + (-5.59 - 8.37i)T + (-27.1 + 65.5i)T^{2}
73 1+(−8.63−5.77i)T+(27.9+67.4i)T2 1 + (-8.63 - 5.77i)T + (27.9 + 67.4i)T^{2}
79 1+(1.25−3.04i)T+(−55.8−55.8i)T2 1 + (1.25 - 3.04i)T + (-55.8 - 55.8i)T^{2}
83 1+(0.917−0.0903i)T+(81.4−16.1i)T2 1 + (0.917 - 0.0903i)T + (81.4 - 16.1i)T^{2}
89 1+(6.22+1.23i)T+(82.2+34.0i)T2 1 + (6.22 + 1.23i)T + (82.2 + 34.0i)T^{2}
97 1+(9.59+9.59i)T+97iT2 1 + (9.59 + 9.59i)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

−11.22390929343920410955335522145, −10.67255027326099615256093728791, −9.531282133572055418663463732269, −8.625034238772635646626421664175, −6.88018937124325631810482526468, −6.36143777730162562192067743707, −5.06696768377045298196722661284, −4.27975799144668538765972430550, −3.13488094841638045860195362219, −1.27350750702614013575895151030, 1.96869811337174984419513361271, 3.69367293231081711887280676495, 4.61704877775419918456229344541, 5.78657049160035168084792881032, 6.69555536542559108118245202835, 7.33552634347908189459188218084, 8.536654191184328817787588521015, 9.660689785883838190516042194626, 11.00543654857144247144752977629, 11.80535506581929034721007432287

Graph of the ZZ-function along the critical line