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Properties

Label 2-799-799.798-c0-0-0
Degree $2$
Conductor $799$
Sign $-0.809 - 0.587i$
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  − 0.618·2-s + 1.17i·3-s − 0.618·4-s − 0.726i·6-s + 1.90i·7-s + 8-s − 0.381·9-s − 0.726i·12-s − 1.17i·14-s + (−0.809 − 0.587i)17-s + 0.236·18-s − 2.23·21-s + 1.17i·24-s − 25-s + 0.726i·27-s − 1.17i·28-s + ⋯
L(s)  = 1  − 0.618·2-s + 1.17i·3-s − 0.618·4-s − 0.726i·6-s + 1.90i·7-s + 8-s − 0.381·9-s − 0.726i·12-s − 1.17i·14-s + (−0.809 − 0.587i)17-s + 0.236·18-s − 2.23·21-s + 1.17i·24-s − 25-s + 0.726i·27-s − 1.17i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.809 - 0.587i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (798, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ -0.809 - 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5437377824\)
\(L(\frac12)\) \(\approx\) \(0.5437377824\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 - T \)
good2 \( 1 + 0.618T + T^{2} \)
3 \( 1 - 1.17iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - 1.90iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.17iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 + 0.618T + T^{2} \)
61 \( 1 + 1.90iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.17iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.17iT - T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 - 1.90iT - T^{2} \)
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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

−10.55659384046715586786129192363, −9.658503490873419471616398030091, −9.230608660697126333432813215695, −8.668012208593368509539302893483, −7.77272380949124463408614760904, −6.30859000117540929748295568381, −5.22943843399799874715578687605, −4.70331704210385831845864675399, −3.51475578958502495483302169726, −2.15833606164075396586359970695, 0.73193377152842458121715506240, 1.85968381314560660724426055721, 3.84099739825948163578429689560, 4.48453321000380256502910651092, 6.01147990756952397556880159451, 7.12765120247226943879114138473, 7.47261228052334532118473597486, 8.249419379799970576825237215070, 9.224021066914751676971607364408, 10.26015650052542518179465895946

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