↑

Properties

Label 2-768-4.3-c2-0-13
Degree 22
Conductor 768768
Sign 11
Analytic cond. 20.926420.9264
Root an. cond. 4.574544.57454
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 7.98·5-s + 2.13i·7-s − 2.99·9-s − 8i·11-s − 11.6·13-s − 13.8i·15-s + 11.8·17-s − 14.9i·19-s − 3.70·21-s − 4.27i·23-s + 38.7·25-s − 5.19i·27-s − 0.573·29-s + 57.4i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.59·5-s + 0.305i·7-s − 0.333·9-s − 0.727i·11-s − 0.898·13-s − 0.921i·15-s + 0.697·17-s − 0.785i·19-s − 0.176·21-s − 0.185i·23-s + 1.54·25-s − 0.192i·27-s − 0.0197·29-s + 1.85i·31-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=( Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+1) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: 11
Analytic conductor: 20.926420.9264
Root analytic conductor: 4.574544.57454
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ768(511,⋅)\chi_{768} (511, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 768, ( :1), 1)(2,\ 768,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.0254567781.025456778
L(12)L(\frac12) ≈\approx 1.0254567781.025456778
L(2)L(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
3 1−1.73iT 1 - 1.73iT
good5 1+7.98T+25T2 1 + 7.98T + 25T^{2}
7 1−2.13iT−49T2 1 - 2.13iT - 49T^{2}
11 1+8iT−121T2 1 + 8iT - 121T^{2}
13 1+11.6T+169T2 1 + 11.6T + 169T^{2}
17 1−11.8T+289T2 1 - 11.8T + 289T^{2}
19 1+14.9iT−361T2 1 + 14.9iT - 361T^{2}
23 1+4.27iT−529T2 1 + 4.27iT - 529T^{2}
29 1+0.573T+841T2 1 + 0.573T + 841T^{2}
31 1−57.4iT−961T2 1 - 57.4iT - 961T^{2}
37 1−27.6T+1.36e3T2 1 - 27.6T + 1.36e3T^{2}
41 1−31.5T+1.68e3T2 1 - 31.5T + 1.68e3T^{2}
43 1−28.7iT−1.84e3T2 1 - 28.7iT - 1.84e3T^{2}
47 1+59.5iT−2.20e3T2 1 + 59.5iT - 2.20e3T^{2}
53 1−31.3T+2.80e3T2 1 - 31.3T + 2.80e3T^{2}
59 1+52.7iT−3.48e3T2 1 + 52.7iT - 3.48e3T^{2}
61 1−59.5T+3.72e3T2 1 - 59.5T + 3.72e3T^{2}
67 1−84.7iT−4.48e3T2 1 - 84.7iT - 4.48e3T^{2}
71 1+42.4iT−5.04e3T2 1 + 42.4iT - 5.04e3T^{2}
73 1−5.42T+5.32e3T2 1 - 5.42T + 5.32e3T^{2}
79 1+44.6iT−6.24e3T2 1 + 44.6iT - 6.24e3T^{2}
83 1+67.7iT−6.88e3T2 1 + 67.7iT - 6.88e3T^{2}
89 1−133.T+7.92e3T2 1 - 133.T + 7.92e3T^{2}
97 1−97.1T+9.40e3T2 1 - 97.1T + 9.40e3T^{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

−10.24126743739568923038195329793, −9.133778896017319828822491654897, −8.433564469040277883159890558352, −7.64713346140218961470575807344, −6.78909535568204383279801060368, −5.43280517766058270849916997483, −4.60832185351745339096050424298, −3.63713204223513886147426660655, −2.79055906189137551391559469443, −0.56175459791356687409147696349, 0.76173144431404712526893897534, 2.42444455581311756825398730990, 3.74323843721123280091270541575, 4.47129934989190525191808470133, 5.71113277820536987021444122758, 6.95873056872034355288707995855, 7.74231349913894543583494049298, 7.895134709727914753901305862285, 9.244659856924934785682893514864, 10.15052448045089152006133839070

Graph of the ZZ-function along the critical line