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Properties

Label 2-2e8-8.5-c7-0-37
Degree 22
Conductor 256256
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 79.970579.9705
Root an. cond. 8.942628.94262
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 84i·3-s − 82i·5-s + 456·7-s − 4.86e3·9-s − 2.52e3i·11-s + 1.07e4i·13-s + 6.88e3·15-s − 1.11e4·17-s − 4.12e3i·19-s + 3.83e4i·21-s − 8.17e4·23-s + 7.14e4·25-s − 2.25e5i·27-s − 9.97e4i·29-s − 4.04e4·31-s + ⋯
L(s)  = 1  + 1.79i·3-s − 0.293i·5-s + 0.502·7-s − 2.22·9-s − 0.571i·11-s + 1.36i·13-s + 0.526·15-s − 0.550·17-s − 0.137i·19-s + 0.902i·21-s − 1.40·23-s + 0.913·25-s − 2.20i·27-s − 0.759i·29-s − 0.244·31-s + ⋯

Functional equation

Λ(s)=(256s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(256s/2 ΓC(s+7/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 256256    =    282^{8}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 79.970579.9705
Root analytic conductor: 8.942628.94262
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ256(129,⋅)\chi_{256} (129, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 256, ( :7/2), 0.707+0.707i)(2,\ 256,\ (\ :7/2),\ 0.707 + 0.707i)

Particular Values

L(4)L(4) ≈\approx 0.67007566480.6700756648
L(12)L(\frac12) ≈\approx 0.67007566480.6700756648
L(92)L(\frac{9}{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 1−84iT−2.18e3T2 1 - 84iT - 2.18e3T^{2}
5 1+82iT−7.81e4T2 1 + 82iT - 7.81e4T^{2}
7 1−456T+8.23e5T2 1 - 456T + 8.23e5T^{2}
11 1+2.52e3iT−1.94e7T2 1 + 2.52e3iT - 1.94e7T^{2}
13 1−1.07e4iT−6.27e7T2 1 - 1.07e4iT - 6.27e7T^{2}
17 1+1.11e4T+4.10e8T2 1 + 1.11e4T + 4.10e8T^{2}
19 1+4.12e3iT−8.93e8T2 1 + 4.12e3iT - 8.93e8T^{2}
23 1+8.17e4T+3.40e9T2 1 + 8.17e4T + 3.40e9T^{2}
29 1+9.97e4iT−1.72e10T2 1 + 9.97e4iT - 1.72e10T^{2}
31 1+4.04e4T+2.75e10T2 1 + 4.04e4T + 2.75e10T^{2}
37 1+4.19e5iT−9.49e10T2 1 + 4.19e5iT - 9.49e10T^{2}
41 1+1.41e5T+1.94e11T2 1 + 1.41e5T + 1.94e11T^{2}
43 1+6.90e5iT−2.71e11T2 1 + 6.90e5iT - 2.71e11T^{2}
47 1+6.82e5T+5.06e11T2 1 + 6.82e5T + 5.06e11T^{2}
53 1−1.81e6iT−1.17e12T2 1 - 1.81e6iT - 1.17e12T^{2}
59 1+9.66e5iT−2.48e12T2 1 + 9.66e5iT - 2.48e12T^{2}
61 1+1.88e6iT−3.14e12T2 1 + 1.88e6iT - 3.14e12T^{2}
67 1+2.96e6iT−6.06e12T2 1 + 2.96e6iT - 6.06e12T^{2}
71 1−2.54e6T+9.09e12T2 1 - 2.54e6T + 9.09e12T^{2}
73 1−1.68e6T+1.10e13T2 1 - 1.68e6T + 1.10e13T^{2}
79 1−4.03e6T+1.92e13T2 1 - 4.03e6T + 1.92e13T^{2}
83 1−5.38e6iT−2.71e13T2 1 - 5.38e6iT - 2.71e13T^{2}
89 1−6.47e6T+4.42e13T2 1 - 6.47e6T + 4.42e13T^{2}
97 1+6.06e6T+8.07e13T2 1 + 6.06e6T + 8.07e13T^{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

−10.72532125753540279067780307607, −9.577656471168676227115989819965, −9.004117622771594297649082735016, −8.105087292105323844963876075342, −6.40310702394613329366114798368, −5.24091705598414179955281739247, −4.40262392439682416425100183962, −3.64416350855205688470013100548, −2.12233151472190567563808185408, −0.16054144682128735812978866171, 1.09218341792945546554962518828, 2.05363110490017659800337975830, 3.12577401834294823773825107298, 5.01206895375848133957518027066, 6.17195982811732043725960572218, 6.98445456768812272635542563173, 7.921895020352833980086120407616, 8.453648364449749936029808142988, 10.04218243679370980611432845593, 11.14925570086343818736714994976

Graph of the ZZ-function along the critical line