Properties

Label 2-799-1.1-c1-0-16
Degree $2$
Conductor $799$
Sign $1$
Analytic cond. $6.38004$
Root an. cond. $2.52587$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.86·2-s + 2.64·3-s + 1.49·4-s − 2.58·5-s − 4.95·6-s + 0.553·7-s + 0.941·8-s + 4.00·9-s + 4.82·10-s + 5.54·11-s + 3.96·12-s + 2.13·13-s − 1.03·14-s − 6.83·15-s − 4.75·16-s + 17-s − 7.49·18-s − 5.27·19-s − 3.86·20-s + 1.46·21-s − 10.3·22-s − 2.77·23-s + 2.49·24-s + 1.67·25-s − 3.99·26-s + 2.66·27-s + 0.827·28-s + ⋯
L(s)  = 1  − 1.32·2-s + 1.52·3-s + 0.748·4-s − 1.15·5-s − 2.02·6-s + 0.209·7-s + 0.332·8-s + 1.33·9-s + 1.52·10-s + 1.67·11-s + 1.14·12-s + 0.592·13-s − 0.276·14-s − 1.76·15-s − 1.18·16-s + 0.242·17-s − 1.76·18-s − 1.20·19-s − 0.864·20-s + 0.319·21-s − 2.21·22-s − 0.578·23-s + 0.508·24-s + 0.334·25-s − 0.783·26-s + 0.513·27-s + 0.156·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.206804561\)
\(L(\frac12)\) \(\approx\) \(1.206804561\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
47 \( 1 + T \)
good2 \( 1 + 1.86T + 2T^{2} \)
3 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 + 2.58T + 5T^{2} \)
7 \( 1 - 0.553T + 7T^{2} \)
11 \( 1 - 5.54T + 11T^{2} \)
13 \( 1 - 2.13T + 13T^{2} \)
19 \( 1 + 5.27T + 19T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 + 3.51T + 31T^{2} \)
37 \( 1 - 4.48T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 8.50T + 43T^{2} \)
53 \( 1 - 3.94T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 7.90T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 4.92T + 73T^{2} \)
79 \( 1 + 3.69T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 8.97T + 89T^{2} \)
97 \( 1 + 9.46T + 97T^{2} \)
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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

−9.880991818885855235609765478119, −9.141000129098511547605476083834, −8.540918728292457865631662310643, −8.095570768808956659108102022616, −7.34140948060597375263484662445, −6.41810294225424577500962720921, −4.17848903421708572597040662827, −3.92058387976827729427153007095, −2.39448404870629072807111914734, −1.10534332824805437005439245374, 1.10534332824805437005439245374, 2.39448404870629072807111914734, 3.92058387976827729427153007095, 4.17848903421708572597040662827, 6.41810294225424577500962720921, 7.34140948060597375263484662445, 8.095570768808956659108102022616, 8.540918728292457865631662310643, 9.141000129098511547605476083834, 9.880991818885855235609765478119

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