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Properties

Label 2-799-799.93-c0-0-3
Degree 22
Conductor 799799
Sign 0.380+0.924i0.380 + 0.924i
Analytic cond. 0.3987520.398752
Root an. cond. 0.6314680.631468
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  + (−0.831 + 0.831i)2-s + (0.763 − 1.84i)3-s − 0.381i·4-s + (0.896 + 2.16i)6-s + (0.431 − 0.178i)7-s + (−0.513 − 0.513i)8-s + (−2.10 − 2.10i)9-s + (−0.703 − 0.291i)12-s + (−0.210 + 0.507i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s + 3.49·18-s − 0.930i·21-s + (−1.33 + 0.554i)24-s + (0.707 + 0.707i)25-s + ⋯
L(s)  = 1  + (−0.831 + 0.831i)2-s + (0.763 − 1.84i)3-s − 0.381i·4-s + (0.896 + 2.16i)6-s + (0.431 − 0.178i)7-s + (−0.513 − 0.513i)8-s + (−2.10 − 2.10i)9-s + (−0.703 − 0.291i)12-s + (−0.210 + 0.507i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s + 3.49·18-s − 0.930i·21-s + (−1.33 + 0.554i)24-s + (0.707 + 0.707i)25-s + ⋯

Functional equation

Λ(s)=(799s/2 ΓC(s) L(s)=((0.380+0.924i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(799s/2 ΓC(s) L(s)=((0.380+0.924i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 799799    =    17⋅4717 \cdot 47
Sign: 0.380+0.924i0.380 + 0.924i
Analytic conductor: 0.3987520.398752
Root analytic conductor: 0.6314680.631468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ799(93,⋅)\chi_{799} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 799, ( :0), 0.380+0.924i)(2,\ 799,\ (\ :0),\ 0.380 + 0.924i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.75637575610.7563757561
L(12)L(\frac12) ≈\approx 0.75637575610.7563757561
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)
bad17 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
47 1+iT 1 + iT
good2 1+(0.831−0.831i)T−iT2 1 + (0.831 - 0.831i)T - iT^{2}
3 1+(−0.763+1.84i)T+(−0.707−0.707i)T2 1 + (-0.763 + 1.84i)T + (-0.707 - 0.707i)T^{2}
5 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
7 1+(−0.431+0.178i)T+(0.707−0.707i)T2 1 + (-0.431 + 0.178i)T + (0.707 - 0.707i)T^{2}
11 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
13 1+T2 1 + T^{2}
19 1+iT2 1 + iT^{2}
23 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
29 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
31 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
37 1+(0.0600−0.144i)T+(−0.707−0.707i)T2 1 + (0.0600 - 0.144i)T + (-0.707 - 0.707i)T^{2}
41 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
43 1−iT2 1 - iT^{2}
53 1+(−1.39+1.39i)T−iT2 1 + (-1.39 + 1.39i)T - iT^{2}
59 1+(−1.14−1.14i)T+iT2 1 + (-1.14 - 1.14i)T + iT^{2}
61 1+(−0.965+0.399i)T+(0.707−0.707i)T2 1 + (-0.965 + 0.399i)T + (0.707 - 0.707i)T^{2}
67 1−T2 1 - T^{2}
71 1+(0.581−1.40i)T+(−0.707−0.707i)T2 1 + (0.581 - 1.40i)T + (-0.707 - 0.707i)T^{2}
73 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
79 1+(0.581+1.40i)T+(−0.707+0.707i)T2 1 + (0.581 + 1.40i)T + (-0.707 + 0.707i)T^{2}
83 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
89 1−1.17iT−T2 1 - 1.17iT - T^{2}
97 1+(−1.57−0.652i)T+(0.707+0.707i)T2 1 + (-1.57 - 0.652i)T + (0.707 + 0.707i)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

−9.894489761527655202307079627021, −8.849007342980339223311596788531, −8.559719564237631442727363349293, −7.60983983031758183388215094648, −7.13408209796798926846644936894, −6.53717174380604937073566680594, −5.44894200584658576524108993443, −3.51776092280825576346281535218, −2.40324343431032421921264042041, −0.983077308087567981076856614977, 2.09975073747178200638069797952, 3.03746057365108343043860821507, 4.08743749776862720098052670117, 5.01365341296172298303341089309, 5.96144313812804438660231382226, 7.84853695305949174057369749948, 8.712281745340990767208863974441, 8.911480319403287142674071878136, 10.03330669101625114048153909436, 10.31096306975160726622657191838

Graph of the ZZ-function along the critical line