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Properties

Label 2-800-5.3-c0-0-2
Degree 22
Conductor 800800
Sign −0.229+0.973i-0.229 + 0.973i
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  + (−1 − i)3-s + (1 − i)7-s + i·9-s − 2·21-s + (−1 − i)23-s − 2i·29-s + (1 + i)43-s + (−1 + i)47-s − i·49-s + (1 + i)63-s + (1 − i)67-s + 2i·69-s + 81-s + (1 + i)83-s + (−2 + 2i)87-s + ⋯
L(s)  = 1  + (−1 − i)3-s + (1 − i)7-s + i·9-s − 2·21-s + (−1 − i)23-s − 2i·29-s + (1 + i)43-s + (−1 + i)47-s − i·49-s + (1 + i)63-s + (1 − i)67-s + 2i·69-s + 81-s + (1 + i)83-s + (−2 + 2i)87-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((−0.229+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s) L(s)=((−0.229+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: −0.229+0.973i-0.229 + 0.973i
Analytic conductor: 0.3992520.399252
Root analytic conductor: 0.6318630.631863
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ800(193,⋅)\chi_{800} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :0), −0.229+0.973i)(2,\ 800,\ (\ :0),\ -0.229 + 0.973i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73164221660.7316422166
L(12)L(\frac12) ≈\approx 0.73164221660.7316422166
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+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
7 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
11 1+T2 1 + T^{2}
13 1+iT2 1 + iT^{2}
17 1−iT2 1 - iT^{2}
19 1−T2 1 - T^{2}
23 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
29 1+2iT−T2 1 + 2iT - T^{2}
31 1+T2 1 + T^{2}
37 1−iT2 1 - iT^{2}
41 1+T2 1 + T^{2}
43 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
47 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
53 1+iT2 1 + iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
71 1+T2 1 + T^{2}
73 1+iT2 1 + iT^{2}
79 1−T2 1 - T^{2}
83 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
89 1−2iT−T2 1 - 2iT - T^{2}
97 1−iT2 1 - iT^{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

−10.52182806991915867942435966588, −9.516285885690007470604571955858, −7.987549958995614787489586635325, −7.79309233068281090636853498539, −6.64426959537030984101543600395, −6.05031302786525693150766574450, −4.90110252422022193942073224709, −4.04655376704796702448425754963, −2.16891100401709272297954391108, −0.912779280286979310418994085698, 1.91087069452013848845759771743, 3.53643054532470293052896772089, 4.68241429286250276903957052486, 5.35118785830813710866323432898, 5.94671590199537883366208087967, 7.23989387664129141346164886590, 8.354335030273642507666476163441, 9.099879670111464566614379913831, 10.03268865522979731250990617072, 10.75221685985421843874095155583

Graph of the ZZ-function along the critical line