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Properties

Label 2-2e7-8.5-c3-0-11
Degree 22
Conductor 128128
Sign −1-1
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  − 6.32i·3-s − 17.8i·5-s − 22.6·7-s − 13.0·9-s + 44.2i·11-s + 17.8i·13-s − 113.·15-s + 70·17-s − 82.2i·19-s + 143. i·21-s − 158.·23-s − 195.·25-s − 88.5i·27-s − 125. i·29-s + 280·33-s + ⋯
L(s)  = 1  − 1.21i·3-s − 1.59i·5-s − 1.22·7-s − 0.481·9-s + 1.21i·11-s + 0.381i·13-s − 1.94·15-s + 0.998·17-s − 0.992i·19-s + 1.48i·21-s − 1.43·23-s − 1.56·25-s − 0.631i·27-s − 0.801i·29-s + 1.47·33-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=(− Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: −1-1
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), −1)(2,\ 128,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) ≈\approx −1.03176i-1.03176i
L(12)L(\frac12) ≈\approx −1.03176i-1.03176i
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+6.32iT−27T2 1 + 6.32iT - 27T^{2}
5 1+17.8iT−125T2 1 + 17.8iT - 125T^{2}
7 1+22.6T+343T2 1 + 22.6T + 343T^{2}
11 1−44.2iT−1.33e3T2 1 - 44.2iT - 1.33e3T^{2}
13 1−17.8iT−2.19e3T2 1 - 17.8iT - 2.19e3T^{2}
17 1−70T+4.91e3T2 1 - 70T + 4.91e3T^{2}
19 1+82.2iT−6.85e3T2 1 + 82.2iT - 6.85e3T^{2}
23 1+158.T+1.21e4T2 1 + 158.T + 1.21e4T^{2}
29 1+125.iT−2.43e4T2 1 + 125. iT - 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1+375.iT−5.06e4T2 1 + 375. iT - 5.06e4T^{2}
41 1+182T+6.89e4T2 1 + 182T + 6.89e4T^{2}
43 1+132.iT−7.95e4T2 1 + 132. iT - 7.95e4T^{2}
47 1−316.T+1.03e5T2 1 - 316.T + 1.03e5T^{2}
53 1−125.iT−1.48e5T2 1 - 125. iT - 1.48e5T^{2}
59 1+82.2iT−2.05e5T2 1 + 82.2iT - 2.05e5T^{2}
61 1−232.iT−2.26e5T2 1 - 232. iT - 2.26e5T^{2}
67 1−221.iT−3.00e5T2 1 - 221. iT - 3.00e5T^{2}
71 1+113.T+3.57e5T2 1 + 113.T + 3.57e5T^{2}
73 1−910T+3.89e5T2 1 - 910T + 3.89e5T^{2}
79 1−678.T+4.93e5T2 1 - 678.T + 4.93e5T^{2}
83 1+714.iT−5.71e5T2 1 + 714. iT - 5.71e5T^{2}
89 1+546T+7.04e5T2 1 + 546T + 7.04e5T^{2}
97 1+490T+9.12e5T2 1 + 490T + 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

−12.40832890459561949129560088840, −12.03581223370086710500124360580, −9.947518204547707664068174667346, −9.179098137008116716528552416196, −7.912089215880290909214454444923, −6.93904359422186065305981708579, −5.71317941436571232451105364763, −4.22586096781788846847025218758, −2.01542653512916390805880529757, −0.52240111283131630807705731875, 3.15706713652802246004115415058, 3.62644832231848533663856288426, 5.70107624936722437144644396843, 6.61216925073114069213018639094, 8.082688529071269249181523519771, 9.694926080346705034738807088031, 10.20748072813757285961104199978, 10.93403934088402298490618874132, 12.19068465020443144312628544609, 13.67899429290759603254175071055

Graph of the ZZ-function along the critical line