↑

Properties

Label 2-1344-24.11-c1-0-21
Degree 22
Conductor 13441344
Sign −i-i
Analytic cond. 10.731810.7318
Root an. cond. 3.275953.27595
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  + (1.22 + 1.22i)3-s + 2.44·5-s + i·7-s + 2.99i·9-s + 2.44i·13-s + (2.99 + 2.99i)15-s + 6i·17-s − 7.34·19-s + (−1.22 + 1.22i)21-s + 0.999·25-s + (−3.67 + 3.67i)27-s + 4.89·29-s − 8i·31-s + 2.44i·35-s + 4.89i·37-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + 1.09·5-s + 0.377i·7-s + 0.999i·9-s + 0.679i·13-s + (0.774 + 0.774i)15-s + 1.45i·17-s − 1.68·19-s + (−0.267 + 0.267i)21-s + 0.199·25-s + (−0.707 + 0.707i)27-s + 0.909·29-s − 1.43i·31-s + 0.414i·35-s + 0.805i·37-s + ⋯

Functional equation

Λ(s)=(1344s/2 ΓC(s) L(s)=(−i Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1344s/2 ΓC(s+1/2) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13441344    =    26⋅3⋅72^{6} \cdot 3 \cdot 7
Sign: −i-i
Analytic conductor: 10.731810.7318
Root analytic conductor: 3.275953.27595
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1344(1247,⋅)\chi_{1344} (1247, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1344, ( :1/2), −i)(2,\ 1344,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) ≈\approx 2.4170653902.417065390
L(12)L(\frac12) ≈\approx 2.4170653902.417065390
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
3 1+(−1.22−1.22i)T 1 + (-1.22 - 1.22i)T
7 1−iT 1 - iT
good5 1−2.44T+5T2 1 - 2.44T + 5T^{2}
11 1−11T2 1 - 11T^{2}
13 1−2.44iT−13T2 1 - 2.44iT - 13T^{2}
17 1−6iT−17T2 1 - 6iT - 17T^{2}
19 1+7.34T+19T2 1 + 7.34T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1−4.89T+29T2 1 - 4.89T + 29T^{2}
31 1+8iT−31T2 1 + 8iT - 31T^{2}
37 1−4.89iT−37T2 1 - 4.89iT - 37T^{2}
41 1+6iT−41T2 1 + 6iT - 41T^{2}
43 1−4.89T+43T2 1 - 4.89T + 43T^{2}
47 1−12T+47T2 1 - 12T + 47T^{2}
53 1−9.79T+53T2 1 - 9.79T + 53T^{2}
59 1−2.44iT−59T2 1 - 2.44iT - 59T^{2}
61 1+2.44iT−61T2 1 + 2.44iT - 61T^{2}
67 1+9.79T+67T2 1 + 9.79T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+8iT−79T2 1 + 8iT - 79T^{2}
83 1−7.34iT−83T2 1 - 7.34iT - 83T^{2}
89 1+18iT−89T2 1 + 18iT - 89T^{2}
97 1−2T+97T2 1 - 2T + 97T^{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

−9.877953973342625506241594554782, −8.797638896315111526209825952057, −8.684696303006516167227922963639, −7.50340461179847692891810876849, −6.25179397249292919492592124636, −5.79666651294002561796875171556, −4.54599955040675263023024731308, −3.88942931954054971697813552874, −2.46220368988041504503211971374, −1.90677320669866998330697495805, 0.905407052361729299583494746936, 2.19896121790962692168036762269, 2.89860909753782316651298128304, 4.18308477610880288272149520959, 5.34552400088163909847411344338, 6.26730928262159633670714046618, 6.95053400429764121749539875574, 7.75944359984751048321146973417, 8.716043919284907837689313091008, 9.243260921690765642757303188088

Graph of the ZZ-function along the critical line