↑

Properties

Label 2-384-24.5-c4-0-39
Degree 22
Conductor 384384
Sign 0.333+0.942i0.333 + 0.942i
Analytic cond. 39.694039.6940
Root an. cond. 6.300326.30032
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 8.48i)3-s − 37.9·5-s + 37.9·7-s + (−62.9 + 50.9i)9-s − 134·11-s + 321. i·13-s + (−113. − 321. i)15-s − 237. i·17-s − 492. i·19-s + (113. + 321. i)21-s + 643. i·23-s + 815·25-s + (−620. − 381. i)27-s − 796.·29-s + 1.10e3·31-s + ⋯
L(s)  = 1  + (0.333 + 0.942i)3-s − 1.51·5-s + 0.774·7-s + (−0.777 + 0.628i)9-s − 1.10·11-s + 1.90i·13-s + (−0.505 − 1.43i)15-s − 0.822i·17-s − 1.36i·19-s + (0.258 + 0.730i)21-s + 1.21i·23-s + 1.30·25-s + (−0.851 − 0.523i)27-s − 0.947·29-s + 1.14·31-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.333+0.942i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+2) L(s)=((0.333+0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.333+0.942i0.333 + 0.942i
Analytic conductor: 39.694039.6940
Root analytic conductor: 6.300326.30032
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ384(65,⋅)\chi_{384} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :2), 0.333+0.942i)(2,\ 384,\ (\ :2),\ 0.333 + 0.942i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.44708957390.4470895739
L(12)L(\frac12) ≈\approx 0.44708957390.4470895739
L(3)L(3) 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
3 1+(−3−8.48i)T 1 + (-3 - 8.48i)T
good5 1+37.9T+625T2 1 + 37.9T + 625T^{2}
7 1−37.9T+2.40e3T2 1 - 37.9T + 2.40e3T^{2}
11 1+134T+1.46e4T2 1 + 134T + 1.46e4T^{2}
13 1−321.iT−2.85e4T2 1 - 321. iT - 2.85e4T^{2}
17 1+237.iT−8.35e4T2 1 + 237. iT - 8.35e4T^{2}
19 1+492.iT−1.30e5T2 1 + 492. iT - 1.30e5T^{2}
23 1−643.iT−2.79e5T2 1 - 643. iT - 2.79e5T^{2}
29 1+796.T+7.07e5T2 1 + 796.T + 7.07e5T^{2}
31 1−1.10e3T+9.23e5T2 1 - 1.10e3T + 9.23e5T^{2}
37 1+1.60e3iT−1.87e6T2 1 + 1.60e3iT - 1.87e6T^{2}
41 1+1.28e3iT−2.82e6T2 1 + 1.28e3iT - 2.82e6T^{2}
43 1−424.iT−3.41e6T2 1 - 424. iT - 3.41e6T^{2}
47 1+2.57e3iT−4.87e6T2 1 + 2.57e3iT - 4.87e6T^{2}
53 1−1.63e3T+7.89e6T2 1 - 1.63e3T + 7.89e6T^{2}
59 1−1.38e3T+1.21e7T2 1 - 1.38e3T + 1.21e7T^{2}
61 1−321.iT−1.38e7T2 1 - 321. iT - 1.38e7T^{2}
67 1+186.iT−2.01e7T2 1 + 186. iT - 2.01e7T^{2}
71 1+1.93e3iT−2.54e7T2 1 + 1.93e3iT - 2.54e7T^{2}
73 1−4.89e3T+2.83e7T2 1 - 4.89e3T + 2.83e7T^{2}
79 1+1.47e3T+3.89e7T2 1 + 1.47e3T + 3.89e7T^{2}
83 1−5.91e3T+4.74e7T2 1 - 5.91e3T + 4.74e7T^{2}
89 1+9.06e3iT−6.27e7T2 1 + 9.06e3iT - 6.27e7T^{2}
97 1−5.85e3T+8.85e7T2 1 - 5.85e3T + 8.85e7T^{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

−10.76483300978353835818409279021, −9.453009859167576410550801797758, −8.742160592726702767474757558586, −7.78332577230208700304679599811, −7.10430480391362138649301682272, −5.19985958850412922096076595441, −4.51948227567807503114102543737, −3.64851983149401973492872741171, −2.31973833912136690193194768659, −0.14272126131083889913452826852, 1.02888819555824534285800994232, 2.66718703119923065581349218555, 3.68890912417119097328602323510, 5.04777850826941420432333828960, 6.20249309461562686878591516958, 7.63010264034747106622935490636, 8.104528871911236134630492685676, 8.302918170253086314552140972825, 10.23536139450935246434569148596, 10.97962159588147097139061176581

Graph of the ZZ-function along the critical line