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Properties

Label 2-2e7-8.5-c3-0-10
Degree 22
Conductor 128128
Sign −0.707−0.707i-0.707 - 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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·3-s + 12i·5-s − 32·7-s − 37·9-s − 8i·11-s + 20i·13-s + 96·15-s − 98·17-s + 88i·19-s + 256i·21-s + 32·23-s − 19·25-s + 80i·27-s − 172i·29-s − 256·31-s + ⋯
L(s)  = 1  − 1.53i·3-s + 1.07i·5-s − 1.72·7-s − 1.37·9-s − 0.219i·11-s + 0.426i·13-s + 1.65·15-s − 1.39·17-s + 1.06i·19-s + 2.66i·21-s + 0.290·23-s − 0.151·25-s + 0.570i·27-s − 1.10i·29-s − 1.48·31-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: −0.707−0.707i-0.707 - 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: 11
Selberg data: (2, 128, ( :3/2), −0.707−0.707i)(2,\ 128,\ (\ :3/2),\ -0.707 - 0.707i)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+8iT−27T2 1 + 8iT - 27T^{2}
5 1−12iT−125T2 1 - 12iT - 125T^{2}
7 1+32T+343T2 1 + 32T + 343T^{2}
11 1+8iT−1.33e3T2 1 + 8iT - 1.33e3T^{2}
13 1−20iT−2.19e3T2 1 - 20iT - 2.19e3T^{2}
17 1+98T+4.91e3T2 1 + 98T + 4.91e3T^{2}
19 1−88iT−6.85e3T2 1 - 88iT - 6.85e3T^{2}
23 1−32T+1.21e4T2 1 - 32T + 1.21e4T^{2}
29 1+172iT−2.43e4T2 1 + 172iT - 2.43e4T^{2}
31 1+256T+2.97e4T2 1 + 256T + 2.97e4T^{2}
37 1−92iT−5.06e4T2 1 - 92iT - 5.06e4T^{2}
41 1+102T+6.89e4T2 1 + 102T + 6.89e4T^{2}
43 1+296iT−7.95e4T2 1 + 296iT - 7.95e4T^{2}
47 1+320T+1.03e5T2 1 + 320T + 1.03e5T^{2}
53 1−76iT−1.48e5T2 1 - 76iT - 1.48e5T^{2}
59 1−408iT−2.05e5T2 1 - 408iT - 2.05e5T^{2}
61 1+636iT−2.26e5T2 1 + 636iT - 2.26e5T^{2}
67 1+552iT−3.00e5T2 1 + 552iT - 3.00e5T^{2}
71 1+416T+3.57e5T2 1 + 416T + 3.57e5T^{2}
73 1+138T+3.89e5T2 1 + 138T + 3.89e5T^{2}
79 1+64T+4.93e5T2 1 + 64T + 4.93e5T^{2}
83 1+392iT−5.71e5T2 1 + 392iT - 5.71e5T^{2}
89 1−582T+7.04e5T2 1 - 582T + 7.04e5T^{2}
97 1−238T+9.12e5T2 1 - 238T + 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

−12.45092757056855816104567684050, −11.37923953594496628618316219292, −10.20760603513653161292808924193, −8.953014507280457733055146880209, −7.48396867514535230660036209520, −6.63439182096268572137556698852, −6.15864662847546624487554837769, −3.46338842835132231093544429303, −2.20797048657158391561318135753, 0, 3.12876406372654968398368489262, 4.35786704626479886346482269382, 5.36702368536121864314502922349, 6.82132329402451852400860869549, 8.903006379956564090130312162996, 9.212338049798465062086091506886, 10.21450542201675788116760123957, 11.20949164116986533597062722664, 12.78634401478974716182020570086

Graph of the ZZ-function along the critical line