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Properties

Label 2-2e8-8.5-c3-0-6
Degree 22
Conductor 256256
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 15.104415.1044
Root an. cond. 3.886443.88644
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  + 8i·3-s − 10i·5-s + 16·7-s − 37·9-s + 40i·11-s + 50i·13-s + 80·15-s − 30·17-s + 40i·19-s + 128i·21-s + 48·23-s + 25·25-s − 80i·27-s + 34i·29-s − 320·31-s + ⋯
L(s)  = 1  + 1.53i·3-s − 0.894i·5-s + 0.863·7-s − 1.37·9-s + 1.09i·11-s + 1.06i·13-s + 1.37·15-s − 0.428·17-s + 0.482i·19-s + 1.33i·21-s + 0.435·23-s + 0.200·25-s − 0.570i·27-s + 0.217i·29-s − 1.85·31-s + ⋯

Functional equation

Λ(s)=(256s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(256s/2 ΓC(s+3/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{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: 256256    =    282^{8}
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 15.104415.1044
Root analytic conductor: 3.886443.88644
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ256(129,⋅)\chi_{256} (129, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 256, ( :3/2), −0.707−0.707i)(2,\ 256,\ (\ :3/2),\ -0.707 - 0.707i)

Particular Values

L(2)L(2) ≈\approx 1.5975752221.597575222
L(12)L(\frac12) ≈\approx 1.5975752221.597575222
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+10iT−125T2 1 + 10iT - 125T^{2}
7 1−16T+343T2 1 - 16T + 343T^{2}
11 1−40iT−1.33e3T2 1 - 40iT - 1.33e3T^{2}
13 1−50iT−2.19e3T2 1 - 50iT - 2.19e3T^{2}
17 1+30T+4.91e3T2 1 + 30T + 4.91e3T^{2}
19 1−40iT−6.85e3T2 1 - 40iT - 6.85e3T^{2}
23 1−48T+1.21e4T2 1 - 48T + 1.21e4T^{2}
29 1−34iT−2.43e4T2 1 - 34iT - 2.43e4T^{2}
31 1+320T+2.97e4T2 1 + 320T + 2.97e4T^{2}
37 1−310iT−5.06e4T2 1 - 310iT - 5.06e4T^{2}
41 1+410T+6.89e4T2 1 + 410T + 6.89e4T^{2}
43 1+152iT−7.95e4T2 1 + 152iT - 7.95e4T^{2}
47 1−416T+1.03e5T2 1 - 416T + 1.03e5T^{2}
53 1+410iT−1.48e5T2 1 + 410iT - 1.48e5T^{2}
59 1−200iT−2.05e5T2 1 - 200iT - 2.05e5T^{2}
61 1+30iT−2.26e5T2 1 + 30iT - 2.26e5T^{2}
67 1−776iT−3.00e5T2 1 - 776iT - 3.00e5T^{2}
71 1−400T+3.57e5T2 1 - 400T + 3.57e5T^{2}
73 1−630T+3.89e5T2 1 - 630T + 3.89e5T^{2}
79 1−1.12e3T+4.93e5T2 1 - 1.12e3T + 4.93e5T^{2}
83 1−552iT−5.71e5T2 1 - 552iT - 5.71e5T^{2}
89 1−326T+7.04e5T2 1 - 326T + 7.04e5T^{2}
97 1+110T+9.12e5T2 1 + 110T + 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

−11.74270911886240281870207657876, −10.90066341406279168920037281994, −9.922421216994580345763145502540, −9.132834781360296760594847863853, −8.443703117772995318659693451716, −7.00336017404331775659543717528, −5.22832895826117415389890931193, −4.72413864921906530323589817852, −3.82277485796557956693962682318, −1.79119022451592537989005078492, 0.63301741996652784432489114438, 2.10056678164313028491213512821, 3.28816367749648520267647967832, 5.33395294850062952231818612913, 6.35509400858744123006377628469, 7.30140909751366380493270886787, 7.987788004469972135605515317038, 8.967081238669298308113551414543, 10.82786542846412693262239802028, 11.07188218745043156842696488615

Graph of the ZZ-function along the critical line