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Properties

Label 2-799-799.610-c0-0-3
Degree $2$
Conductor $799$
Sign $0.997 + 0.0758i$
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  + (0.292 + 0.707i)3-s − i·4-s + (0.707 + 0.292i)7-s + (0.292 − 0.292i)9-s + (0.707 − 0.292i)12-s − 16-s − 17-s + 0.585i·21-s + (0.707 − 0.707i)25-s + (1.00 + 0.414i)27-s + (0.292 − 0.707i)28-s + (−0.292 − 0.292i)36-s + (0.707 + 1.70i)37-s + i·47-s + (−0.292 − 0.707i)48-s + (−0.292 − 0.292i)49-s + ⋯
L(s)  = 1  + (0.292 + 0.707i)3-s − i·4-s + (0.707 + 0.292i)7-s + (0.292 − 0.292i)9-s + (0.707 − 0.292i)12-s − 16-s − 17-s + 0.585i·21-s + (0.707 − 0.707i)25-s + (1.00 + 0.414i)27-s + (0.292 − 0.707i)28-s + (−0.292 − 0.292i)36-s + (0.707 + 1.70i)37-s + i·47-s + (−0.292 − 0.707i)48-s + (−0.292 − 0.292i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.997 + 0.0758i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (610, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 0.997 + 0.0758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144715778\)
\(L(\frac12)\) \(\approx\) \(1.144715778\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + T \)
47 \( 1 - iT \)
good2 \( 1 + iT^{2} \)
3 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
61 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.43359193067705147938585120747, −9.608944251214677514615236098992, −8.986359899023080206480232186033, −8.131894431859671650806072049790, −6.81267419863250867388740139920, −6.08019481742297483042924822901, −4.79876274307482899583782422644, −4.46965917503732683491478775816, −2.90150301558418022506540329796, −1.52141779790946051256389488615, 1.74253400638481136706800819564, 2.81681354362779972551415009849, 4.11512688321308378648967089916, 4.91331005623260117295370210204, 6.42239838291363031223512022359, 7.32592420002085790698621608933, 7.76807304728872810474613486327, 8.609641524747135311054981874582, 9.376179755129705291301232878869, 10.82126890142242602561803433222

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