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Properties

Label 2-2e7-8.5-c3-0-6
Degree 22
Conductor 128128
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 7.552247.55224
Root an. cond. 2.748132.74813
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·5-s + 27·9-s − 92i·13-s + 94·17-s + 109·25-s − 284i·29-s + 396i·37-s − 230·41-s − 108i·45-s − 343·49-s + 572i·53-s + 468i·61-s − 368·65-s − 1.09e3·73-s + 729·81-s + ⋯
L(s)  = 1  − 0.357i·5-s + 9-s − 1.96i·13-s + 1.34·17-s + 0.871·25-s − 1.81i·29-s + 1.75i·37-s − 0.876·41-s − 0.357i·45-s − 49-s + 1.48i·53-s + 0.982i·61-s − 0.702·65-s − 1.76·73-s + 0.999·81-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{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: 128128    =    272^{7}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 7.552247.55224
Root analytic conductor: 2.748132.74813
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ128(65,⋅)\chi_{128} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :3/2), 0.707+0.707i)(2,\ 128,\ (\ :3/2),\ 0.707 + 0.707i)

Particular Values

L(2)L(2) ≈\approx 1.57612−0.652850i1.57612 - 0.652850i
L(12)L(\frac12) ≈\approx 1.57612−0.652850i1.57612 - 0.652850i
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
good3 1−27T2 1 - 27T^{2}
5 1+4iT−125T2 1 + 4iT - 125T^{2}
7 1+343T2 1 + 343T^{2}
11 1−1.33e3T2 1 - 1.33e3T^{2}
13 1+92iT−2.19e3T2 1 + 92iT - 2.19e3T^{2}
17 1−94T+4.91e3T2 1 - 94T + 4.91e3T^{2}
19 1−6.85e3T2 1 - 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1+284iT−2.43e4T2 1 + 284iT - 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1−396iT−5.06e4T2 1 - 396iT - 5.06e4T^{2}
41 1+230T+6.89e4T2 1 + 230T + 6.89e4T^{2}
43 1−7.95e4T2 1 - 7.95e4T^{2}
47 1+1.03e5T2 1 + 1.03e5T^{2}
53 1−572iT−1.48e5T2 1 - 572iT - 1.48e5T^{2}
59 1−2.05e5T2 1 - 2.05e5T^{2}
61 1−468iT−2.26e5T2 1 - 468iT - 2.26e5T^{2}
67 1−3.00e5T2 1 - 3.00e5T^{2}
71 1+3.57e5T2 1 + 3.57e5T^{2}
73 1+1.09e3T+3.89e5T2 1 + 1.09e3T + 3.89e5T^{2}
79 1+4.93e5T2 1 + 4.93e5T^{2}
83 1−5.71e5T2 1 - 5.71e5T^{2}
89 1−1.67e3T+7.04e5T2 1 - 1.67e3T + 7.04e5T^{2}
97 1+594T+9.12e5T2 1 + 594T + 9.12e5T^{2}
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   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

−12.78484732429561978649617370521, −11.90764113632520561511652251883, −10.40303101307729163173774827611, −9.852509159261334332860912531266, −8.309045790252605057850777275456, −7.48428172509871663596900133783, −5.94473080992444743154752892682, −4.74578903472573108525590239224, −3.15780485715801365597521995324, −1.02517492739886578989439019968, 1.65573843803848776582738571451, 3.60300395994520040456512855161, 4.94421065723907476703441997808, 6.60777374400933069046152240068, 7.36912480561218050578042127831, 8.896593904414148534767087127391, 9.867588944926950366979321077115, 10.91463585366786182503598448256, 12.02125191394609853393009838453, 12.90736782961264201904883385231

Graph of the ZZ-function along the critical line