Properties

Label 2-819-91.23-c1-0-5
Degree $2$
Conductor $819$
Sign $-0.280 + 0.959i$
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  + 2.22i·2-s − 2.93·4-s + (0.701 + 0.404i)5-s + (−2.44 − 1.00i)7-s − 2.08i·8-s + (−0.900 + 1.55i)10-s + (2.66 + 1.53i)11-s + (−3.01 + 1.97i)13-s + (2.24 − 5.43i)14-s − 1.23·16-s − 2.79·17-s + (−3.73 + 2.15i)19-s + (−2.06 − 1.19i)20-s + (−3.41 + 5.92i)22-s − 4.95·23-s + ⋯
L(s)  = 1  + 1.57i·2-s − 1.46·4-s + (0.313 + 0.181i)5-s + (−0.924 − 0.381i)7-s − 0.738i·8-s + (−0.284 + 0.492i)10-s + (0.803 + 0.463i)11-s + (−0.836 + 0.548i)13-s + (0.599 − 1.45i)14-s − 0.309·16-s − 0.678·17-s + (−0.856 + 0.494i)19-s + (−0.461 − 0.266i)20-s + (−0.729 + 1.26i)22-s − 1.03·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.280 + 0.959i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.329871 - 0.439954i\)
\(L(\frac12)\) \(\approx\) \(0.329871 - 0.439954i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.44 + 1.00i)T \)
13 \( 1 + (3.01 - 1.97i)T \)
good2 \( 1 - 2.22iT - 2T^{2} \)
5 \( 1 + (-0.701 - 0.404i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.66 - 1.53i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.79T + 17T^{2} \)
19 \( 1 + (3.73 - 2.15i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.95T + 23T^{2} \)
29 \( 1 + (-2.84 - 4.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.93 + 1.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.72iT - 37T^{2} \)
41 \( 1 + (8.48 - 4.90i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.85 - 4.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.31 - 3.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.83 + 4.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.93iT - 59T^{2} \)
61 \( 1 + (-1.51 - 2.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.59 + 4.96i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.84 - 4.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.75 - 1.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.13 + 1.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.52iT - 83T^{2} \)
89 \( 1 - 3.32iT - 89T^{2} \)
97 \( 1 + (-11.0 - 6.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.48105983018393565997723412875, −9.705849401372641534384767858307, −9.037744690113965564601726777221, −8.093670457621379252067826955149, −7.13160879552923775806058336863, −6.52855589311405999981292784471, −6.03084052380651656679648235857, −4.70573352542406640230433246524, −3.96062453400462737596572928363, −2.23358870705329788024744009262, 0.25333972238892839996631815147, 1.89826343171287628928915826645, 2.85093625750291024080902320972, 3.78870426372495260500094194732, 4.80992403272321365333503882238, 6.06657862647870705083880145302, 6.88768490261522388938596262567, 8.435913955647121020212725258093, 9.091092948672731763197824649613, 9.938640740311489189928739947341

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