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Properties

Label 2-799-17.16-c1-0-21
Degree $2$
Conductor $799$
Sign $0.659 - 0.751i$
Analytic cond. $6.38004$
Root an. cond. $2.52587$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.67·2-s − 1.20i·3-s + 5.14·4-s + 1.97i·5-s + 3.23i·6-s + 5.12i·7-s − 8.39·8-s + 1.53·9-s − 5.27i·10-s − 3.26i·11-s − 6.22i·12-s + 6.27·13-s − 13.7i·14-s + 2.38·15-s + 12.1·16-s + (2.71 − 3.10i)17-s + ⋯
L(s)  = 1  − 1.88·2-s − 0.698i·3-s + 2.57·4-s + 0.883i·5-s + 1.31i·6-s + 1.93i·7-s − 2.96·8-s + 0.512·9-s − 1.66i·10-s − 0.984i·11-s − 1.79i·12-s + 1.74·13-s − 3.66i·14-s + 0.617·15-s + 3.03·16-s + (0.659 − 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.659 - 0.751i$
Analytic conductor: \(6.38004\)
Root analytic conductor: \(2.52587\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.659324 + 0.298826i\)
\(L(\frac12)\) \(\approx\) \(0.659324 + 0.298826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.71 + 3.10i)T \)
47 \( 1 - T \)
good2 \( 1 + 2.67T + 2T^{2} \)
3 \( 1 + 1.20iT - 3T^{2} \)
5 \( 1 - 1.97iT - 5T^{2} \)
7 \( 1 - 5.12iT - 7T^{2} \)
11 \( 1 + 3.26iT - 11T^{2} \)
13 \( 1 - 6.27T + 13T^{2} \)
19 \( 1 + 6.12T + 19T^{2} \)
23 \( 1 - 3.97iT - 23T^{2} \)
29 \( 1 - 3.75iT - 29T^{2} \)
31 \( 1 + 2.38iT - 31T^{2} \)
37 \( 1 - 6.07iT - 37T^{2} \)
41 \( 1 - 0.399iT - 41T^{2} \)
43 \( 1 - 9.49T + 43T^{2} \)
53 \( 1 - 2.70T + 53T^{2} \)
59 \( 1 + 2.20T + 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 + 2.26T + 67T^{2} \)
71 \( 1 + 5.70iT - 71T^{2} \)
73 \( 1 - 5.66iT - 73T^{2} \)
79 \( 1 + 1.26iT - 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 - 0.0910iT - 97T^{2} \)
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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

−10.33974051558524916702357402675, −9.259032482271328405501109208472, −8.675881851132224499389633248256, −8.106210290321843056060087860472, −7.11511759386502733214263962755, −6.22212192081612757762524739685, −5.88166829596589273282484014896, −3.20896466633798583443407880534, −2.36303410215982106095567517158, −1.25175886359651539636735424788, 0.803019940040413012867923860533, 1.68417871721350639497571772937, 3.78204913515857641011608752857, 4.44229325176746078375939670691, 6.20816209084376398339766637499, 7.02737357931051601324202786772, 7.84672279979879990283754930669, 8.579318557228721916097287627058, 9.340377538917224899278838787385, 10.23020839567118671224273233202

Graph of the $Z$-function along the critical line