↑

Properties

Label 2-384-128.109-c1-0-22
Degree 22
Conductor 384384
Sign 0.189+0.981i0.189 + 0.981i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.208 − 1.39i)2-s + (0.881 + 0.471i)3-s + (−1.91 − 0.583i)4-s + (1.48 − 1.81i)5-s + (0.843 − 1.13i)6-s + (2.68 + 1.79i)7-s + (−1.21 + 2.55i)8-s + (0.555 + 0.831i)9-s + (−2.22 − 2.46i)10-s + (−0.671 − 0.203i)11-s + (−1.41 − 1.41i)12-s + (4.24 − 3.48i)13-s + (3.06 − 3.37i)14-s + (2.16 − 0.897i)15-s + (3.31 + 2.23i)16-s + (−4.31 − 1.78i)17-s + ⋯
L(s)  = 1  + (0.147 − 0.989i)2-s + (0.509 + 0.272i)3-s + (−0.956 − 0.291i)4-s + (0.665 − 0.810i)5-s + (0.344 − 0.463i)6-s + (1.01 + 0.677i)7-s + (−0.429 + 0.902i)8-s + (0.185 + 0.277i)9-s + (−0.703 − 0.777i)10-s + (−0.202 − 0.0613i)11-s + (−0.407 − 0.408i)12-s + (1.17 − 0.966i)13-s + (0.819 − 0.902i)14-s + (0.559 − 0.231i)15-s + (0.829 + 0.558i)16-s + (−1.04 − 0.433i)17-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.189+0.981i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=((0.189+0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.189+0.981i0.189 + 0.981i
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ384(109,⋅)\chi_{384} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), 0.189+0.981i)(2,\ 384,\ (\ :1/2),\ 0.189 + 0.981i)

Particular Values

L(1)L(1) ≈\approx 1.43767−1.18691i1.43767 - 1.18691i
L(12)L(\frac12) ≈\approx 1.43767−1.18691i1.43767 - 1.18691i
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+(−0.208+1.39i)T 1 + (-0.208 + 1.39i)T
3 1+(−0.881−0.471i)T 1 + (-0.881 - 0.471i)T
good5 1+(−1.48+1.81i)T+(−0.975−4.90i)T2 1 + (-1.48 + 1.81i)T + (-0.975 - 4.90i)T^{2}
7 1+(−2.68−1.79i)T+(2.67+6.46i)T2 1 + (-2.68 - 1.79i)T + (2.67 + 6.46i)T^{2}
11 1+(0.671+0.203i)T+(9.14+6.11i)T2 1 + (0.671 + 0.203i)T + (9.14 + 6.11i)T^{2}
13 1+(−4.24+3.48i)T+(2.53−12.7i)T2 1 + (-4.24 + 3.48i)T + (2.53 - 12.7i)T^{2}
17 1+(4.31+1.78i)T+(12.0+12.0i)T2 1 + (4.31 + 1.78i)T + (12.0 + 12.0i)T^{2}
19 1+(0.0391−0.397i)T+(−18.6−3.70i)T2 1 + (0.0391 - 0.397i)T + (-18.6 - 3.70i)T^{2}
23 1+(2.48+0.493i)T+(21.2+8.80i)T2 1 + (2.48 + 0.493i)T + (21.2 + 8.80i)T^{2}
29 1+(0.449+1.48i)T+(−24.1+16.1i)T2 1 + (0.449 + 1.48i)T + (-24.1 + 16.1i)T^{2}
31 1+(3.23−3.23i)T−31iT2 1 + (3.23 - 3.23i)T - 31iT^{2}
37 1+(6.20−0.610i)T+(36.2−7.21i)T2 1 + (6.20 - 0.610i)T + (36.2 - 7.21i)T^{2}
41 1+(−1.22+6.16i)T+(−37.8−15.6i)T2 1 + (-1.22 + 6.16i)T + (-37.8 - 15.6i)T^{2}
43 1+(−10.2+5.46i)T+(23.8−35.7i)T2 1 + (-10.2 + 5.46i)T + (23.8 - 35.7i)T^{2}
47 1+(3.46−8.36i)T+(−33.2−33.2i)T2 1 + (3.46 - 8.36i)T + (-33.2 - 33.2i)T^{2}
53 1+(2.45−8.09i)T+(−44.0−29.4i)T2 1 + (2.45 - 8.09i)T + (-44.0 - 29.4i)T^{2}
59 1+(−2.49−2.04i)T+(11.5+57.8i)T2 1 + (-2.49 - 2.04i)T + (11.5 + 57.8i)T^{2}
61 1+(5.58−10.4i)T+(−33.8−50.7i)T2 1 + (5.58 - 10.4i)T + (-33.8 - 50.7i)T^{2}
67 1+(6.84−12.8i)T+(−37.2−55.7i)T2 1 + (6.84 - 12.8i)T + (-37.2 - 55.7i)T^{2}
71 1+(−7.22+10.8i)T+(−27.1−65.5i)T2 1 + (-7.22 + 10.8i)T + (-27.1 - 65.5i)T^{2}
73 1+(4.10−2.74i)T+(27.9−67.4i)T2 1 + (4.10 - 2.74i)T + (27.9 - 67.4i)T^{2}
79 1+(−3.49−8.42i)T+(−55.8+55.8i)T2 1 + (-3.49 - 8.42i)T + (-55.8 + 55.8i)T^{2}
83 1+(−2.18−0.215i)T+(81.4+16.1i)T2 1 + (-2.18 - 0.215i)T + (81.4 + 16.1i)T^{2}
89 1+(0.00810−0.00161i)T+(82.2−34.0i)T2 1 + (0.00810 - 0.00161i)T + (82.2 - 34.0i)T^{2}
97 1+(−2.67+2.67i)T−97iT2 1 + (-2.67 + 2.67i)T - 97iT^{2}
show more
show less
   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

−11.00023672527215181418323612398, −10.41868349752258103226732388701, −9.009381781812360122700672740994, −8.905029663690181280872102223498, −7.88549259299796070871337832306, −5.79128718695626271893366889980, −5.13445313434952978737350299041, −4.04527997053083765582383986303, −2.59424953665795233192429836215, −1.46348744475586084042342821716, 1.86592134860658508485805416463, 3.66610014389958856177648580508, 4.66474861560179859820704936458, 6.12671623983782729093842523636, 6.75912331925927530997555317835, 7.73711492255292472208412177207, 8.558514305712053142842933187298, 9.456569355153846198333776309828, 10.59775278101764319642885406258, 11.38424495416166485934362490744

Graph of the ZZ-function along the critical line