Properties

Label 2-2e8-8.5-c7-0-5
Degree 22
Conductor 256256
Sign 0.7070.707i-0.707 - 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  − 58i·5-s + 2.18e3·9-s + 8.89e3i·13-s − 4.00e4·17-s + 7.47e4·25-s − 2.33e5i·29-s + 5.63e5i·37-s − 9.53e3·41-s − 1.26e5i·45-s − 8.23e5·49-s − 7.98e5i·53-s + 3.50e6i·61-s + 5.16e5·65-s − 3.91e6·73-s + 4.78e6·81-s + ⋯
L(s)  = 1  − 0.207i·5-s + 9-s + 1.12i·13-s − 1.97·17-s + 0.956·25-s − 1.77i·29-s + 1.83i·37-s − 0.0215·41-s − 0.207i·45-s − 49-s − 0.736i·53-s + 1.97i·61-s + 0.233·65-s − 1.17·73-s + 81-s + ⋯

Functional equation

Λ(s)=(256s/2ΓC(s)L(s)=((0.7070.707i)Λ(8s)\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/2ΓC(s+7/2)L(s)=((0.7070.707i)Λ(1s)\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: 0.7070.707i-0.707 - 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), 0.7070.707i)(2,\ 256,\ (\ :7/2),\ -0.707 - 0.707i)

Particular Values

L(4)L(4) \approx 0.85880773090.8588077309
L(12)L(\frac12) \approx 0.85880773090.8588077309
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
good3 12.18e3T2 1 - 2.18e3T^{2}
5 1+58iT7.81e4T2 1 + 58iT - 7.81e4T^{2}
7 1+8.23e5T2 1 + 8.23e5T^{2}
11 11.94e7T2 1 - 1.94e7T^{2}
13 18.89e3iT6.27e7T2 1 - 8.89e3iT - 6.27e7T^{2}
17 1+4.00e4T+4.10e8T2 1 + 4.00e4T + 4.10e8T^{2}
19 18.93e8T2 1 - 8.93e8T^{2}
23 1+3.40e9T2 1 + 3.40e9T^{2}
29 1+2.33e5iT1.72e10T2 1 + 2.33e5iT - 1.72e10T^{2}
31 1+2.75e10T2 1 + 2.75e10T^{2}
37 15.63e5iT9.49e10T2 1 - 5.63e5iT - 9.49e10T^{2}
41 1+9.53e3T+1.94e11T2 1 + 9.53e3T + 1.94e11T^{2}
43 12.71e11T2 1 - 2.71e11T^{2}
47 1+5.06e11T2 1 + 5.06e11T^{2}
53 1+7.98e5iT1.17e12T2 1 + 7.98e5iT - 1.17e12T^{2}
59 12.48e12T2 1 - 2.48e12T^{2}
61 13.50e6iT3.14e12T2 1 - 3.50e6iT - 3.14e12T^{2}
67 16.06e12T2 1 - 6.06e12T^{2}
71 1+9.09e12T2 1 + 9.09e12T^{2}
73 1+3.91e6T+1.10e13T2 1 + 3.91e6T + 1.10e13T^{2}
79 1+1.92e13T2 1 + 1.92e13T^{2}
83 12.71e13T2 1 - 2.71e13T^{2}
89 1+9.24e6T+4.42e13T2 1 + 9.24e6T + 4.42e13T^{2}
97 1+1.75e7T+8.07e13T2 1 + 1.75e7T + 8.07e13T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.20255807738571982125331653307, −10.10120348455262132812530448503, −9.229796821030971709167218716401, −8.323826884162768501970937251353, −7.01468541366574405450022340293, −6.38757839404935632178326071947, −4.73182339802749230894809410282, −4.14090996317143924029891382775, −2.43809156404570826930909614831, −1.32489436017951333899372613098, 0.19462660925242371691595585269, 1.59889341212149281161137643746, 2.90568590634306187181472596953, 4.19144476908664730311459225371, 5.22514309247016007453366243162, 6.59003793280927384686564897775, 7.31380105152670964941081467558, 8.527168978966713563428536029372, 9.441281789637447849265232830041, 10.62613301583465156320378916628

Graph of the ZZ-function along the critical line