Properties

Label 2-799-17.16-c1-0-59
Degree $2$
Conductor $799$
Sign $-0.618 + 0.785i$
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  + 1.53·2-s − 2.41i·3-s + 0.362·4-s + 2.19i·5-s − 3.71i·6-s − 3.09i·7-s − 2.51·8-s − 2.84·9-s + 3.37i·10-s − 5.28i·11-s − 0.877i·12-s + 5.92·13-s − 4.76i·14-s + 5.31·15-s − 4.59·16-s + (−2.54 + 3.24i)17-s + ⋯
L(s)  = 1  + 1.08·2-s − 1.39i·3-s + 0.181·4-s + 0.982i·5-s − 1.51i·6-s − 1.17i·7-s − 0.889·8-s − 0.948·9-s + 1.06i·10-s − 1.59i·11-s − 0.253i·12-s + 1.64·13-s − 1.27i·14-s + 1.37·15-s − 1.14·16-s + (−0.618 + 0.785i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.618 + 0.785i$
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: \((2,\ 799,\ (\ :1/2),\ -0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908141 - 1.87029i\)
\(L(\frac12)\) \(\approx\) \(0.908141 - 1.87029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.54 - 3.24i)T \)
47 \( 1 - T \)
good2 \( 1 - 1.53T + 2T^{2} \)
3 \( 1 + 2.41iT - 3T^{2} \)
5 \( 1 - 2.19iT - 5T^{2} \)
7 \( 1 + 3.09iT - 7T^{2} \)
11 \( 1 + 5.28iT - 11T^{2} \)
13 \( 1 - 5.92T + 13T^{2} \)
19 \( 1 + 7.76T + 19T^{2} \)
23 \( 1 + 5.08iT - 23T^{2} \)
29 \( 1 + 7.62iT - 29T^{2} \)
31 \( 1 - 6.15iT - 31T^{2} \)
37 \( 1 + 3.84iT - 37T^{2} \)
41 \( 1 - 7.98iT - 41T^{2} \)
43 \( 1 + 2.02T + 43T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 6.23T + 59T^{2} \)
61 \( 1 - 1.73iT - 61T^{2} \)
67 \( 1 - 8.01T + 67T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 + 0.762iT - 73T^{2} \)
79 \( 1 + 7.27iT - 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 7.14T + 89T^{2} \)
97 \( 1 - 0.772iT - 97T^{2} \)
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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

−10.52511729157179061256935985121, −8.550687136657694751263707106559, −8.350489047645644157703207657651, −6.90501198539423748113103931723, −6.34666971192952910323342857018, −6.01792290193378023129247450453, −4.21450379715340614332208511844, −3.57059224184123773062432526129, −2.41154665291404236939126640244, −0.73664281234543018180827539997, 2.19593205538277792547914329308, 3.66218589011972675803803473841, 4.33913649470262072982617231244, 5.04322562705913128460729912249, 5.61359093415647736734180090646, 6.72585203919869402665963035655, 8.545493116273301636825744986475, 8.969981424668046665978556842826, 9.504394105806429752289045451639, 10.61750014935328676817192988400

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