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Properties

Label 2-819-91.41-c1-0-34
Degree $2$
Conductor $819$
Sign $-0.837 - 0.545i$
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  + (−1.61 − 0.433i)2-s + (0.700 + 0.404i)4-s + (−1.42 − 1.42i)5-s + (−1.11 − 2.39i)7-s + (1.41 + 1.41i)8-s + (1.68 + 2.91i)10-s + (−0.254 + 0.948i)11-s + (1.60 − 3.22i)13-s + (0.764 + 4.36i)14-s + (−2.48 − 4.29i)16-s + (2.99 − 5.18i)17-s + (−2.71 + 0.726i)19-s + (−0.421 − 1.57i)20-s + (0.822 − 1.42i)22-s + (−2.58 + 1.49i)23-s + ⋯
L(s)  = 1  + (−1.14 − 0.306i)2-s + (0.350 + 0.202i)4-s + (−0.635 − 0.635i)5-s + (−0.421 − 0.906i)7-s + (0.498 + 0.498i)8-s + (0.532 + 0.922i)10-s + (−0.0766 + 0.285i)11-s + (0.446 − 0.894i)13-s + (0.204 + 1.16i)14-s + (−0.620 − 1.07i)16-s + (0.725 − 1.25i)17-s + (−0.622 + 0.166i)19-s + (−0.0941 − 0.351i)20-s + (0.175 − 0.303i)22-s + (−0.539 + 0.311i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.837 - 0.545i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.837 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0701113 + 0.236132i\)
\(L(\frac12)\) \(\approx\) \(0.0701113 + 0.236132i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 2.39i)T \)
13 \( 1 + (-1.60 + 3.22i)T \)
good2 \( 1 + (1.61 + 0.433i)T + (1.73 + i)T^{2} \)
5 \( 1 + (1.42 + 1.42i)T + 5iT^{2} \)
11 \( 1 + (0.254 - 0.948i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.99 + 5.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.71 - 0.726i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.58 - 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.65 + 6.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.11 - 6.11i)T + 31iT^{2} \)
37 \( 1 + (1.24 - 4.63i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.886 - 3.30i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.748 + 0.432i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.17 - 2.17i)T - 47iT^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + (0.131 + 0.491i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (11.2 + 6.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.827 + 0.221i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.01 - 11.2i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.03 + 1.03i)T - 73iT^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (7.75 + 2.07i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.0 + 3.23i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.917693907795188815618351042560, −8.925026562918618064436417117645, −7.951743197465965089616571007033, −7.72749151734869635420405749833, −6.47860057591925919432539759322, −5.10832904873950764759288165805, −4.24935435164093292816190850907, −2.98645329367287873521385247538, −1.25730400491476861283310603447, −0.20155579681702041113669439146, 1.79916526271141933959997031025, 3.33825054508524037653317810944, 4.25072683228871287107671503638, 5.86823784614050524511707132330, 6.59080623851309446405367418403, 7.50435296195529474291678112602, 8.339592689315838180515160031295, 8.897137080690759737233787524999, 9.732331339854181438165388876344, 10.60735744407762669516043085743

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