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Properties

Label 2-800-40.19-c0-0-0
Degree 22
Conductor 800800
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 0.3992520.399252
Root an. cond. 0.6318630.631863
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·3-s + 11-s + i·17-s − 19-s + i·27-s + i·33-s − 41-s − 2i·43-s − 49-s − 51-s − i·57-s + 2·59-s − i·67-s − i·73-s − 81-s + ⋯
L(s)  = 1  + i·3-s + 11-s + i·17-s − 19-s + i·27-s + i·33-s − 41-s − 2i·43-s − 49-s − 51-s − i·57-s + 2·59-s − i·67-s − i·73-s − 81-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 0.3992520.399252
Root analytic conductor: 0.6318630.631863
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ800(399,⋅)\chi_{800} (399, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :0), 0.447−0.894i)(2,\ 800,\ (\ :0),\ 0.447 - 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0240871241.024087124
L(12)L(\frac12) ≈\approx 1.0240871241.024087124
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 1
good3 1−iT−T2 1 - iT - T^{2}
7 1+T2 1 + T^{2}
11 1−T+T2 1 - T + T^{2}
13 1+T2 1 + T^{2}
17 1−iT−T2 1 - iT - T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+T2 1 + T^{2}
29 1−T2 1 - T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1+T+T2 1 + T + T^{2}
43 1+2iT−T2 1 + 2iT - T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + T^{2}
59 1−2T+T2 1 - 2T + T^{2}
61 1−T2 1 - T^{2}
67 1+iT−T2 1 + iT - T^{2}
71 1−T2 1 - T^{2}
73 1+iT−T2 1 + iT - T^{2}
79 1−T2 1 - T^{2}
83 1−iT−T2 1 - iT - T^{2}
89 1−T+T2 1 - T + T^{2}
97 1+2iT−T2 1 + 2iT - 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

−10.48192462057179431669655267948, −9.888970010120026042719453466201, −8.974148495807358145771356534967, −8.360859442106513473343411294213, −7.05938144605083630950688159385, −6.24893824838174372412763924405, −5.13329225274981955757854630763, −4.14552487292280802101351485904, −3.54888068522024869069231136553, −1.84068117969644956727528700491, 1.27683073765345448383954374058, 2.49032747626348307530879777677, 3.89058780516858376452091421139, 4.96042355551600179416120835091, 6.30808991530566763121933921838, 6.76979535282880298690896417891, 7.66976433374811804530926445176, 8.524658810398930347786710734400, 9.433726876346548985088267965846, 10.26471179692433523487791927714

Graph of the ZZ-function along the critical line