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Properties

Label 2-819-91.59-c1-0-40
Degree $2$
Conductor $819$
Sign $-0.975 - 0.219i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  + (−0.698 − 0.698i)2-s − 1.02i·4-s + (0.912 + 0.244i)5-s + (2.61 + 0.407i)7-s + (−2.11 + 2.11i)8-s + (−0.466 − 0.808i)10-s + (−6.35 − 1.70i)11-s + (−3.43 + 1.08i)13-s + (−1.54 − 2.11i)14-s + 0.904·16-s − 3.73·17-s + (−0.325 − 1.21i)19-s + (0.250 − 0.933i)20-s + (3.24 + 5.62i)22-s − 0.233i·23-s + ⋯
L(s)  = 1  + (−0.494 − 0.494i)2-s − 0.511i·4-s + (0.407 + 0.109i)5-s + (0.988 + 0.154i)7-s + (−0.746 + 0.746i)8-s + (−0.147 − 0.255i)10-s + (−1.91 − 0.513i)11-s + (−0.953 + 0.299i)13-s + (−0.411 − 0.564i)14-s + 0.226·16-s − 0.905·17-s + (−0.0747 − 0.278i)19-s + (0.0559 − 0.208i)20-s + (0.692 + 1.19i)22-s − 0.0486i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.975 - 0.219i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.975 - 0.219i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0474970 + 0.427592i\)
\(L(\frac12)\) \(\approx\) \(0.0474970 + 0.427592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.61 - 0.407i)T \)
13 \( 1 + (3.43 - 1.08i)T \)
good2 \( 1 + (0.698 + 0.698i)T + 2iT^{2} \)
5 \( 1 + (-0.912 - 0.244i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (6.35 + 1.70i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + (0.325 + 1.21i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.233iT - 23T^{2} \)
29 \( 1 + (-2.32 + 4.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.96 + 7.35i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.55 + 3.55i)T - 37iT^{2} \)
41 \( 1 + (2.49 + 9.29i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (10.4 - 6.00i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.563 + 2.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.04 - 3.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.27 - 5.27i)T + 59iT^{2} \)
61 \( 1 + (2.12 + 1.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.63 - 9.82i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.433 - 1.61i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.85 - 2.10i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.942 + 1.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.95 + 9.95i)T - 83iT^{2} \)
89 \( 1 + (-2.88 - 2.88i)T + 89iT^{2} \)
97 \( 1 + (-2.41 - 0.647i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.01461017754258200705426162918, −9.025910358778885608249429665780, −8.225357178481370566474561047205, −7.44274113393776157266328491024, −6.06361860106226830944600424488, −5.33975543047497066115046380383, −4.53742337215321932588044325601, −2.54654232651134361293627044705, −2.10528043039621713538603933720, −0.22447587621633110328618538988, 2.01653468806632679773366332648, 3.12838989573712561673126241458, 4.72335600847743617290529590333, 5.24425335292442747369111336727, 6.62152910101595449085627183559, 7.51761859636080830743555623827, 8.023897872985876543117669835256, 8.763797900793968357089826671975, 9.857344972246420393242331500261, 10.43969698660428206781040117797

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