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Properties

Label 2-1280-40.19-c0-0-0
Degree 22
Conductor 12801280
Sign 0.707−0.707i0.707 - 0.707i
Analytic cond. 0.6388030.638803
Root an. cond. 0.7992510.799251
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s + 9-s − 25-s + 2i·29-s + 2·41-s + i·45-s − 49-s − 2i·61-s + 81-s − 2·89-s − 2i·101-s − 2i·109-s + ⋯
L(s)  = 1  + i·5-s + 9-s − 25-s + 2i·29-s + 2·41-s + i·45-s − 49-s − 2i·61-s + 81-s − 2·89-s − 2i·101-s − 2i·109-s + ⋯

Functional equation

Λ(s)=(1280s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: 0.707−0.707i0.707 - 0.707i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1280(639,⋅)\chi_{1280} (639, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1280, ( :0), 0.707−0.707i)(2,\ 1280,\ (\ :0),\ 0.707 - 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1317345781.131734578
L(12)L(\frac12) ≈\approx 1.1317345781.131734578
L(1)L(1) 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
5 1−iT 1 - iT
good3 1−T2 1 - T^{2}
7 1+T2 1 + T^{2}
11 1+T2 1 + T^{2}
13 1+T2 1 + T^{2}
17 1−T2 1 - T^{2}
19 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1−2T+T2 1 - 2T + T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + T^{2}
59 1+T2 1 + T^{2}
61 1+2iT−T2 1 + 2iT - T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+2T+T2 1 + 2T + T^{2}
97 1−T2 1 - T^{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

−9.952380009061971476016893250330, −9.362415640710904540750174732075, −8.246088991136250399594171153945, −7.33608252330103242552435231391, −6.83982491707451663541364549824, −5.93330150882845923892834145932, −4.82352186882573125275769387637, −3.82502464749591129704786151542, −2.89733515607916253952647196850, −1.64330031213782616999717196204, 1.12985572546244192274549107844, 2.38980190813354691460753359718, 3.97423021975834343415153124354, 4.50307621547822240437210255771, 5.54359395567518135212106836468, 6.39194979146782392669591705024, 7.51041393700793237534338313363, 8.064006775761831723103436634520, 9.081758883693930174934303337483, 9.667459165055266968595816369154

Graph of the ZZ-function along the critical line