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Properties

Label 2-799-799.93-c0-0-3
Degree $2$
Conductor $799$
Sign $0.380 + 0.924i$
Analytic cond. $0.398752$
Root an. cond. $0.631468$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.380 + 0.924i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 0.380 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7563757561\)
\(L(\frac12)\) \(\approx\) \(0.7563757561\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + iT \)
good2 \( 1 + (0.831 - 0.831i)T - iT^{2} \)
3 \( 1 + (-0.763 + 1.84i)T + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.431 + 0.178i)T + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.0600 - 0.144i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
59 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
61 \( 1 + (-0.965 + 0.399i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.581 - 1.40i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.581 + 1.40i)T + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 - 1.17iT - T^{2} \)
97 \( 1 + (-1.57 - 0.652i)T + (0.707 + 0.707i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line