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Properties

Label 2-819-63.59-c1-0-88
Degree $2$
Conductor $819$
Sign $0.427 + 0.904i$
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  + (1.78 + 1.03i)2-s + (0.518 − 1.65i)3-s + (1.12 + 1.95i)4-s − 3.13·5-s + (2.63 − 2.41i)6-s + (1.93 − 1.80i)7-s + 0.526i·8-s + (−2.46 − 1.71i)9-s + (−5.59 − 3.23i)10-s − 3.36i·11-s + (3.81 − 0.850i)12-s + (0.866 + 0.5i)13-s + (5.32 − 1.21i)14-s + (−1.62 + 5.17i)15-s + (1.71 − 2.96i)16-s + (−1.26 + 2.18i)17-s + ⋯
L(s)  = 1  + (1.26 + 0.729i)2-s + (0.299 − 0.954i)3-s + (0.563 + 0.976i)4-s − 1.40·5-s + (1.07 − 0.986i)6-s + (0.732 − 0.680i)7-s + 0.186i·8-s + (−0.820 − 0.571i)9-s + (−1.76 − 1.02i)10-s − 1.01i·11-s + (1.10 − 0.245i)12-s + (0.240 + 0.138i)13-s + (1.42 − 0.325i)14-s + (−0.419 + 1.33i)15-s + (0.427 − 0.741i)16-s + (−0.306 + 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.427 + 0.904i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.427 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12630 - 1.34715i\)
\(L(\frac12)\) \(\approx\) \(2.12630 - 1.34715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.518 + 1.65i)T \)
7 \( 1 + (-1.93 + 1.80i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (-1.78 - 1.03i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.13T + 5T^{2} \)
11 \( 1 + 3.36iT - 11T^{2} \)
17 \( 1 + (1.26 - 2.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.58 - 1.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.03iT - 23T^{2} \)
29 \( 1 + (-2.78 + 1.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.72 + 5.03i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.32 + 2.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.80 - 3.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.01 - 1.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.32 - 9.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.63 - 1.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.14 - 8.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.8 - 7.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.09 - 8.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.09iT - 71T^{2} \)
73 \( 1 + (5.29 + 3.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.79 - 3.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.27 + 7.41i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.43 + 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.14 + 4.70i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.33636249774268935267010743600, −8.467781745039066166346791949550, −8.223149041591185810929264112997, −7.37314509998757514162371128196, −6.58303679537476421783127737129, −5.85127747922738463886552902062, −4.43295927919858828731323298989, −3.97931720735535034622322740886, −2.84434966091128973042711528944, −0.825329198955374972629994876769, 2.15218689506141872995892282926, 3.21584338243663158021282641029, 4.03368713408277550684216917468, 4.81752683740796310739131025660, 5.25061219474790332431654187742, 6.82408595930162713078845276454, 8.148244966407869655555390638387, 8.524973798252364897952166912027, 9.783044891496082639799794809848, 10.73923792905356378032514083243

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