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Properties

Label 2-799-17.16-c1-0-3
Degree $2$
Conductor $799$
Sign $-0.888 + 0.458i$
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  + 1.38·2-s + 3.44i·3-s − 0.0681·4-s + 1.16i·5-s + 4.79i·6-s − 3.53i·7-s − 2.87·8-s − 8.88·9-s + 1.62i·10-s − 0.450i·11-s − 0.235i·12-s − 2.66·13-s − 4.91i·14-s − 4.02·15-s − 3.85·16-s + (−3.66 + 1.89i)17-s + ⋯
L(s)  = 1  + 0.982·2-s + 1.99i·3-s − 0.0340·4-s + 0.521i·5-s + 1.95i·6-s − 1.33i·7-s − 1.01·8-s − 2.96·9-s + 0.512i·10-s − 0.135i·11-s − 0.0678i·12-s − 0.738·13-s − 1.31i·14-s − 1.03·15-s − 0.964·16-s + (−0.888 + 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.888 + 0.458i$
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: \((2,\ 799,\ (\ :1/2),\ -0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240303 - 0.990059i\)
\(L(\frac12)\) \(\approx\) \(0.240303 - 0.990059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.66 - 1.89i)T \)
47 \( 1 - T \)
good2 \( 1 - 1.38T + 2T^{2} \)
3 \( 1 - 3.44iT - 3T^{2} \)
5 \( 1 - 1.16iT - 5T^{2} \)
7 \( 1 + 3.53iT - 7T^{2} \)
11 \( 1 + 0.450iT - 11T^{2} \)
13 \( 1 + 2.66T + 13T^{2} \)
19 \( 1 - 0.0470T + 19T^{2} \)
23 \( 1 - 5.06iT - 23T^{2} \)
29 \( 1 - 10.2iT - 29T^{2} \)
31 \( 1 - 4.16iT - 31T^{2} \)
37 \( 1 - 3.13iT - 37T^{2} \)
41 \( 1 - 3.94iT - 41T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
53 \( 1 + 8.99T + 53T^{2} \)
59 \( 1 - 2.10T + 59T^{2} \)
61 \( 1 + 2.20iT - 61T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 - 0.791iT - 71T^{2} \)
73 \( 1 + 9.32iT - 73T^{2} \)
79 \( 1 + 1.27iT - 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + 2.44iT - 97T^{2} \)
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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

−10.79111768478010730452414658994, −10.04635841885557027199873574294, −9.284086337557638225942960174561, −8.506463119965136933889396112310, −7.09514971018175405564754768524, −6.01530478242903023744874951532, −4.91881396876881868981303650629, −4.53934633405977466939094657639, −3.54958041071260100747056049377, −3.05606055812905653234412984699, 0.34547551756806743238341643585, 2.26249708333351499584451136834, 2.70795700437507903544742065899, 4.54415675480944504817700450291, 5.53994967391498651345601814487, 6.10445332309511054392752740492, 6.95398604678973361184366224601, 8.042872353873500736547795884207, 8.762720938263379534769630163032, 9.351852417133171218735689458079

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