Properties

Label 1-799-799.798-r1-0-0
Degree $1$
Conductor $799$
Sign $1$
Analytic cond. $85.8644$
Root an. cond. $85.8644$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 13-s − 14-s − 15-s + 16-s + 18-s − 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 29-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 13-s − 14-s − 15-s + 16-s + 18-s − 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 29-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $1$
Analytic conductor: \(85.8644\)
Root analytic conductor: \(85.8644\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{799} (798, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 799,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.680239380\)
\(L(\frac12)\) \(\approx\) \(3.680239380\)
\(L(1)\) \(\approx\) \(1.778264938\)
\(L(1)\) \(\approx\) \(1.778264938\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
47 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.14611491562738857665527104464, −21.56020999829903406115169098750, −20.89722233442326965992572647680, −19.56092360818553434527292041603, −19.14961411520427231038571579220, −17.73899463820179802018638101587, −16.93886181922339955527803191826, −16.65135822155825556859606788477, −15.53575603439418420959744242664, −14.72860689846592601785899482196, −13.76823012122303927929501289847, −12.97825702769493271297154491247, −12.36783065798973622667635184231, −11.65765808519263789778538522513, −10.46507258850699937617689300449, −10.04884492055929502266477127324, −8.9630635784036319424900597296, −7.17175868808639979403998810499, −6.56581281552915242379169263140, −6.0558214058541419353478397620, −5.06068100087980954555952835912, −4.31584178938478777844501951630, −3.07801006932293405663369041662, −2.034176685075387365369085723789, −0.86009735932971197832404787463, 0.86009735932971197832404787463, 2.034176685075387365369085723789, 3.07801006932293405663369041662, 4.31584178938478777844501951630, 5.06068100087980954555952835912, 6.0558214058541419353478397620, 6.56581281552915242379169263140, 7.17175868808639979403998810499, 8.9630635784036319424900597296, 10.04884492055929502266477127324, 10.46507258850699937617689300449, 11.65765808519263789778538522513, 12.36783065798973622667635184231, 12.97825702769493271297154491247, 13.76823012122303927929501289847, 14.72860689846592601785899482196, 15.53575603439418420959744242664, 16.65135822155825556859606788477, 16.93886181922339955527803191826, 17.73899463820179802018638101587, 19.14961411520427231038571579220, 19.56092360818553434527292041603, 20.89722233442326965992572647680, 21.56020999829903406115169098750, 22.14611491562738857665527104464

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