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Properties

Label 2-819-91.89-c1-0-11
Degree $2$
Conductor $819$
Sign $-0.999 + 0.00241i$
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  + (1.79 + 1.79i)2-s + 4.44i·4-s + (0.590 + 2.20i)5-s + (−2.51 + 0.826i)7-s + (−4.38 + 4.38i)8-s + (−2.89 + 5.02i)10-s + (0.449 + 1.67i)11-s + (−1.05 − 3.44i)13-s + (−5.99 − 3.02i)14-s − 6.86·16-s + 7.51·17-s + (−5.80 − 1.55i)19-s + (−9.80 + 2.62i)20-s + (−2.20 + 3.81i)22-s + 0.0216i·23-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)2-s + 2.22i·4-s + (0.264 + 0.986i)5-s + (−0.949 + 0.312i)7-s + (−1.55 + 1.55i)8-s + (−0.916 + 1.58i)10-s + (0.135 + 0.505i)11-s + (−0.292 − 0.956i)13-s + (−1.60 − 0.809i)14-s − 1.71·16-s + 1.82·17-s + (−1.33 − 0.356i)19-s + (−2.19 + 0.587i)20-s + (−0.469 + 0.813i)22-s + 0.00451i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.999 + 0.00241i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.999 + 0.00241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00326372 - 2.70065i\)
\(L(\frac12)\) \(\approx\) \(0.00326372 - 2.70065i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 - 0.826i)T \)
13 \( 1 + (1.05 + 3.44i)T \)
good2 \( 1 + (-1.79 - 1.79i)T + 2iT^{2} \)
5 \( 1 + (-0.590 - 2.20i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.449 - 1.67i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 7.51T + 17T^{2} \)
19 \( 1 + (5.80 + 1.55i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.0216iT - 23T^{2} \)
29 \( 1 + (0.432 + 0.749i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.81 - 1.55i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.64 - 6.64i)T - 37iT^{2} \)
41 \( 1 + (-2.45 - 0.656i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.71 - 0.987i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.62 - 1.50i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.87 - 11.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.10 - 4.10i)T + 59iT^{2} \)
61 \( 1 + (-3.12 + 1.80i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.08 + 0.290i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (12.5 - 3.36i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.56 + 9.58i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.16 + 3.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.64 + 8.64i)T - 83iT^{2} \)
89 \( 1 + (1.71 + 1.71i)T + 89iT^{2} \)
97 \( 1 + (1.99 + 7.43i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.47445070460910450106016639824, −9.958113802474740581505490401062, −8.623830275032090139628787131036, −7.68248123441277223762714552117, −6.94034515195512842786437845886, −6.25040471281355456216197365070, −5.62091169077027783130850451043, −4.54880564715344862317424876868, −3.32591311178395954357987758622, −2.77631388509620145664450179362, 0.910689041292568240348607414254, 2.15769077209968629894142518792, 3.44628855199732548219695176011, 4.11015086264009197478934504261, 5.15744769419250342155987061299, 5.88040918868928847294983749503, 6.82570117862606140815041644899, 8.370775737632104471126277370222, 9.388850872741367904281286166442, 10.00396170212589548278541833158

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