↑

Properties

Label 2-1152-8.5-c3-0-38
Degree 22
Conductor 11521152
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6i·5-s + 21.1·7-s − 42.3i·11-s + 20i·13-s − 8·17-s + 84.6i·19-s + 169.·23-s + 89·25-s − 46i·29-s − 21.1·31-s − 126. i·35-s + 164i·37-s + 312·41-s − 423. i·43-s − 169.·47-s + ⋯
L(s)  = 1  − 0.536i·5-s + 1.14·7-s − 1.16i·11-s + 0.426i·13-s − 0.114·17-s + 1.02i·19-s + 1.53·23-s + 0.711·25-s − 0.294i·29-s − 0.122·31-s − 0.613i·35-s + 0.728i·37-s + 1.18·41-s − 1.50i·43-s − 0.525·47-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1152(577,⋅)\chi_{1152} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), 0.707+0.707i)(2,\ 1152,\ (\ :3/2),\ 0.707 + 0.707i)

Particular Values

L(2)L(2) ≈\approx 2.5673754652.567375465
L(12)L(\frac12) ≈\approx 2.5673754652.567375465
L(52)L(\frac{5}{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+6iT−125T2 1 + 6iT - 125T^{2}
7 1−21.1T+343T2 1 - 21.1T + 343T^{2}
11 1+42.3iT−1.33e3T2 1 + 42.3iT - 1.33e3T^{2}
13 1−20iT−2.19e3T2 1 - 20iT - 2.19e3T^{2}
17 1+8T+4.91e3T2 1 + 8T + 4.91e3T^{2}
19 1−84.6iT−6.85e3T2 1 - 84.6iT - 6.85e3T^{2}
23 1−169.T+1.21e4T2 1 - 169.T + 1.21e4T^{2}
29 1+46iT−2.43e4T2 1 + 46iT - 2.43e4T^{2}
31 1+21.1T+2.97e4T2 1 + 21.1T + 2.97e4T^{2}
37 1−164iT−5.06e4T2 1 - 164iT - 5.06e4T^{2}
41 1−312T+6.89e4T2 1 - 312T + 6.89e4T^{2}
43 1+423.iT−7.95e4T2 1 + 423. iT - 7.95e4T^{2}
47 1+169.T+1.03e5T2 1 + 169.T + 1.03e5T^{2}
53 1+266iT−1.48e5T2 1 + 266iT - 1.48e5T^{2}
59 1−253.iT−2.05e5T2 1 - 253. iT - 2.05e5T^{2}
61 1+132iT−2.26e5T2 1 + 132iT - 2.26e5T^{2}
67 1−507.iT−3.00e5T2 1 - 507. iT - 3.00e5T^{2}
71 1+677.T+3.57e5T2 1 + 677.T + 3.57e5T^{2}
73 1+246T+3.89e5T2 1 + 246T + 3.89e5T^{2}
79 1−232.T+4.93e5T2 1 - 232.T + 4.93e5T^{2}
83 1−973.iT−5.71e5T2 1 - 973. iT - 5.71e5T^{2}
89 1−1.39e3T+7.04e5T2 1 - 1.39e3T + 7.04e5T^{2}
97 1+302T+9.12e5T2 1 + 302T + 9.12e5T^{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

−9.032959481127559728152602678245, −8.579720491216612238277776743613, −7.84444744088947442729253045814, −6.86247919410912032856713318743, −5.77377340618477275039220994881, −5.05403635175349150163910024257, −4.18917185355927322950393102587, −3.04778070124184201568645686863, −1.68938816589699488749758997620, −0.76254989645313024606813594334, 1.01512033486156650575088197395, 2.20643780495507675980572054619, 3.15862577324839819390455785523, 4.63306543984598574957539009257, 4.94998026026571007814556191358, 6.26186390420236299262986898798, 7.23686578783479364261199594819, 7.65418099098983203744304805846, 8.788701521465357771514201068547, 9.428835738369168115366466927328

Graph of the ZZ-function along the critical line