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Properties

Label 2-799-17.16-c1-0-55
Degree $2$
Conductor $799$
Sign $-0.242 + 0.970i$
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-s + 2i·3-s − 4-s − 2i·5-s + 2i·6-s − 2i·7-s − 3·8-s − 9-s − 2i·10-s − 2i·12-s − 6·13-s − 2i·14-s + 4·15-s − 16-s + (1 − 4i)17-s − 18-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15i·3-s − 0.5·4-s − 0.894i·5-s + 0.816i·6-s − 0.755i·7-s − 1.06·8-s − 0.333·9-s − 0.632i·10-s − 0.577i·12-s − 1.66·13-s − 0.534i·14-s + 1.03·15-s − 0.250·16-s + (0.242 − 0.970i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.242 + 0.970i$
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: \((2,\ 799,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509657 - 0.652756i\)
\(L(\frac12)\) \(\approx\) \(0.509657 - 0.652756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1 + 4i)T \)
47 \( 1 + T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 14iT - 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.11271094414282697652952933199, −9.184118889646668991650418880523, −8.668019094075134371647838523088, −7.45012238839212978773672684395, −6.25046919592258909536482575893, −4.81321834501475260915730228166, −4.77396604622088489306436562174, −4.03603393227382306558746763330, −2.68830776627265719885343016017, −0.31463588881634273490302445604, 1.97956658395530429860310232864, 2.90246821007771644564779512378, 4.13760420084956340092608838135, 5.31787434362844414545756436806, 6.15317460658504527352723156143, 6.93140024102843105477719463152, 7.76696905702186188089203852192, 8.705722897166589843326537748555, 9.650454349130134573495672387669, 10.57972015967534394254111295986

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