↑

Properties

Label 2-24e2-8.3-c2-0-4
Degree 22
Conductor 576576
Sign 0.258−0.965i0.258 - 0.965i
Analytic cond. 15.694815.6948
Root an. cond. 3.961673.96167
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  − 3.46i·5-s + 10i·7-s + 13.8·11-s + 6.92i·13-s − 30·17-s − 6.92·19-s − 12i·23-s + 13.0·25-s + 51.9i·29-s + 14i·31-s + 34.6·35-s + 55.4i·37-s + 6·41-s + 62.3·43-s + 84i·47-s + ⋯
L(s)  = 1  − 0.692i·5-s + 1.42i·7-s + 1.25·11-s + 0.532i·13-s − 1.76·17-s − 0.364·19-s − 0.521i·23-s + 0.520·25-s + 1.79i·29-s + 0.451i·31-s + 0.989·35-s + 1.49i·37-s + 0.146·41-s + 1.45·43-s + 1.78i·47-s + ⋯

Functional equation

Λ(s)=(576s/2 ΓC(s) L(s)=((0.258−0.965i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(576s/2 ΓC(s+1) L(s)=((0.258−0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 576576    =    26⋅322^{6} \cdot 3^{2}
Sign: 0.258−0.965i0.258 - 0.965i
Analytic conductor: 15.694815.6948
Root analytic conductor: 3.961673.96167
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ576(415,⋅)\chi_{576} (415, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 576, ( :1), 0.258−0.965i)(2,\ 576,\ (\ :1),\ 0.258 - 0.965i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.5417209461.541720946
L(12)L(\frac12) ≈\approx 1.5417209461.541720946
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
good5 1+3.46iT−25T2 1 + 3.46iT - 25T^{2}
7 1−10iT−49T2 1 - 10iT - 49T^{2}
11 1−13.8T+121T2 1 - 13.8T + 121T^{2}
13 1−6.92iT−169T2 1 - 6.92iT - 169T^{2}
17 1+30T+289T2 1 + 30T + 289T^{2}
19 1+6.92T+361T2 1 + 6.92T + 361T^{2}
23 1+12iT−529T2 1 + 12iT - 529T^{2}
29 1−51.9iT−841T2 1 - 51.9iT - 841T^{2}
31 1−14iT−961T2 1 - 14iT - 961T^{2}
37 1−55.4iT−1.36e3T2 1 - 55.4iT - 1.36e3T^{2}
41 1−6T+1.68e3T2 1 - 6T + 1.68e3T^{2}
43 1−62.3T+1.84e3T2 1 - 62.3T + 1.84e3T^{2}
47 1−84iT−2.20e3T2 1 - 84iT - 2.20e3T^{2}
53 1−17.3iT−2.80e3T2 1 - 17.3iT - 2.80e3T^{2}
59 1−62.3T+3.48e3T2 1 - 62.3T + 3.48e3T^{2}
61 1+96.9iT−3.72e3T2 1 + 96.9iT - 3.72e3T^{2}
67 1−48.4T+4.48e3T2 1 - 48.4T + 4.48e3T^{2}
71 1−60iT−5.04e3T2 1 - 60iT - 5.04e3T^{2}
73 1+86T+5.32e3T2 1 + 86T + 5.32e3T^{2}
79 1−38iT−6.24e3T2 1 - 38iT - 6.24e3T^{2}
83 1+13.8T+6.88e3T2 1 + 13.8T + 6.88e3T^{2}
89 1+78T+7.92e3T2 1 + 78T + 7.92e3T^{2}
97 1−62T+9.40e3T2 1 - 62T + 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

−10.90224244428363022207449041198, −9.453411902735921273701493323607, −8.859263750765588976718936343055, −8.532101527460893961976831888999, −6.86218345965042272201075070055, −6.24655490795604291462452845787, −5.03844751664607348139678014881, −4.24792991917049444724925928247, −2.69489393388727772336867656866, −1.47183051499872596619843961828, 0.61283480987577486908157251457, 2.27809206313197820495624340122, 3.81276735275334532379363049809, 4.31880862200366076178991506554, 5.96676204210569846680043446906, 6.86905918201796305504872498717, 7.38893709189746923848017209327, 8.606816162038943483748287018081, 9.552019412062116224604496767825, 10.49028524096563118541734104410

Graph of the ZZ-function along the critical line