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Properties

Label 2-799-799.798-c0-0-10
Degree $2$
Conductor $799$
Sign $0.309 + 0.951i$
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  + 1.61·2-s − 1.90i·3-s + 1.61·4-s − 3.07i·6-s + 1.17i·7-s + 8-s − 2.61·9-s − 3.07i·12-s + 1.90i·14-s + (0.309 + 0.951i)17-s − 4.23·18-s + 2.23·21-s − 1.90i·24-s − 25-s + 3.07i·27-s + 1.90i·28-s + ⋯
L(s)  = 1  + 1.61·2-s − 1.90i·3-s + 1.61·4-s − 3.07i·6-s + 1.17i·7-s + 8-s − 2.61·9-s − 3.07i·12-s + 1.90i·14-s + (0.309 + 0.951i)17-s − 4.23·18-s + 2.23·21-s − 1.90i·24-s − 25-s + 3.07i·27-s + 1.90i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.309 + 0.951i$
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: \((2,\ 799,\ (\ :0),\ 0.309 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103670194\)
\(L(\frac12)\) \(\approx\) \(2.103670194\)
\(L(1)\) 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 \( 1 - T \)
good2 \( 1 - 1.61T + T^{2} \)
3 \( 1 + 1.90iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - 1.17iT - T^{2} \)
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 \( 1 + 1.90iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 - 1.61T + T^{2} \)
61 \( 1 + 1.17iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.90iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.90iT - T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 - 1.17iT - T^{2} \)
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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

−10.92730154303691863188511160998, −9.191939069420361291852238577934, −8.273583310429530167253117940190, −7.44494324473914114134352655656, −6.49927892204442259880984748444, −5.86076824090073874896331840795, −5.36466029016716930557473677270, −3.74545548360107902941522006729, −2.57705585961564947296051439386, −1.89740679322066617615386798472, 2.81711902218600475804408072674, 3.64964892789936085296986540618, 4.30614247235089430614961612031, 4.97094963084410077150673851390, 5.75878655530612182249045911703, 6.82735316295868090658360247170, 8.041441912268850933251627090273, 9.306409840874296298853476323030, 10.00773479758399957644495840651, 10.75542627810894982901629047608

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