↑

Properties

Label 2-1152-12.11-c3-0-19
Degree 22
Conductor 11521152
Sign 0.816−0.577i0.816 - 0.577i
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  + 9.77i·5-s − 16.6i·7-s − 12.3·11-s + 74.0·13-s − 18.4i·17-s + 106. i·19-s − 116.·23-s + 29.4·25-s − 93.6i·29-s + 158. i·31-s + 162.·35-s + 50.7·37-s − 315. i·41-s − 254. i·43-s − 355.·47-s + ⋯
L(s)  = 1  + 0.874i·5-s − 0.896i·7-s − 0.338·11-s + 1.57·13-s − 0.262i·17-s + 1.28i·19-s − 1.05·23-s + 0.235·25-s − 0.599i·29-s + 0.916i·31-s + 0.784·35-s + 0.225·37-s − 1.20i·41-s − 0.903i·43-s − 1.10·47-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=((0.816−0.577i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=((0.816−0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 0.816−0.577i0.816 - 0.577i
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1152(1151,⋅)\chi_{1152} (1151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), 0.816−0.577i)(2,\ 1152,\ (\ :3/2),\ 0.816 - 0.577i)

Particular Values

L(2)L(2) ≈\approx 2.1014344652.101434465
L(12)L(\frac12) ≈\approx 2.1014344652.101434465
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 1−9.77iT−125T2 1 - 9.77iT - 125T^{2}
7 1+16.6iT−343T2 1 + 16.6iT - 343T^{2}
11 1+12.3T+1.33e3T2 1 + 12.3T + 1.33e3T^{2}
13 1−74.0T+2.19e3T2 1 - 74.0T + 2.19e3T^{2}
17 1+18.4iT−4.91e3T2 1 + 18.4iT - 4.91e3T^{2}
19 1−106.iT−6.85e3T2 1 - 106. iT - 6.85e3T^{2}
23 1+116.T+1.21e4T2 1 + 116.T + 1.21e4T^{2}
29 1+93.6iT−2.43e4T2 1 + 93.6iT - 2.43e4T^{2}
31 1−158.iT−2.97e4T2 1 - 158. iT - 2.97e4T^{2}
37 1−50.7T+5.06e4T2 1 - 50.7T + 5.06e4T^{2}
41 1+315.iT−6.89e4T2 1 + 315. iT - 6.89e4T^{2}
43 1+254.iT−7.95e4T2 1 + 254. iT - 7.95e4T^{2}
47 1+355.T+1.03e5T2 1 + 355.T + 1.03e5T^{2}
53 1−150.iT−1.48e5T2 1 - 150. iT - 1.48e5T^{2}
59 1−504.T+2.05e5T2 1 - 504.T + 2.05e5T^{2}
61 1−281.T+2.26e5T2 1 - 281.T + 2.26e5T^{2}
67 1+101.iT−3.00e5T2 1 + 101. iT - 3.00e5T^{2}
71 1−797.T+3.57e5T2 1 - 797.T + 3.57e5T^{2}
73 1−1.00e3T+3.89e5T2 1 - 1.00e3T + 3.89e5T^{2}
79 1−189.iT−4.93e5T2 1 - 189. iT - 4.93e5T^{2}
83 1+475.T+5.71e5T2 1 + 475.T + 5.71e5T^{2}
89 1−1.52e3iT−7.04e5T2 1 - 1.52e3iT - 7.04e5T^{2}
97 1−700.T+9.12e5T2 1 - 700.T + 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

−9.700659583078965074485101677445, −8.501206198310666941873567143867, −7.88214424640161828076930385064, −6.93823434067807208934492416293, −6.28727392530114298182842749003, −5.35753330915335745791999856884, −3.92613720059093326079185016995, −3.52583735655678375933260974395, −2.14603317122278167024385987889, −0.876419348344388113585903907126, 0.66867542158694727101592620528, 1.84084583119309565337861685554, 3.02728488983373063779265158073, 4.19317671211415101079716278478, 5.10797618753897388396651936318, 5.90456904979894808725034240049, 6.67317875183414525446911084362, 8.041806844418248117699285033420, 8.515352574074020633847258322359, 9.189085454639054682787450644286

Graph of the ZZ-function along the critical line