↑

Properties

Label 2-819-91.25-c1-0-18
Degree $2$
Conductor $819$
Sign $0.795 - 0.606i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.929 − 0.536i)2-s + (−0.424 + 0.734i)4-s + (−0.541 + 0.312i)5-s + (2.34 − 1.21i)7-s + 3.05i·8-s + (−0.335 + 0.581i)10-s + (0.613 + 0.354i)11-s + (0.848 + 3.50i)13-s + (1.53 − 2.39i)14-s + (0.791 + 1.37i)16-s + (1.67 − 2.89i)17-s + (−4.50 + 2.60i)19-s − 0.530i·20-s + 0.760·22-s + (2.21 + 3.83i)23-s + ⋯
L(s)  = 1  + (0.657 − 0.379i)2-s + (−0.212 + 0.367i)4-s + (−0.242 + 0.139i)5-s + (0.888 − 0.459i)7-s + 1.08i·8-s + (−0.106 + 0.183i)10-s + (0.185 + 0.106i)11-s + (0.235 + 0.971i)13-s + (0.409 − 0.638i)14-s + (0.197 + 0.342i)16-s + (0.405 − 0.702i)17-s + (−1.03 + 0.596i)19-s − 0.118i·20-s + 0.162·22-s + (0.462 + 0.800i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.795 - 0.606i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.795 - 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96845 + 0.665002i\)
\(L(\frac12)\) \(\approx\) \(1.96845 + 0.665002i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.34 + 1.21i)T \)
13 \( 1 + (-0.848 - 3.50i)T \)
good2 \( 1 + (-0.929 + 0.536i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.541 - 0.312i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.613 - 0.354i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.67 + 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.50 - 2.60i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 - 3.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.59T + 29T^{2} \)
31 \( 1 + (-3.80 - 2.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.366 - 0.211i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (-6.99 + 4.03i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.348 - 0.603i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.54 + 4.93i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.34 + 4.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.02 + 5.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0iT - 71T^{2} \)
73 \( 1 + (4.40 + 2.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.95 - 3.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (11.5 - 6.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.202iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.62862948062428541121031645996, −9.433230048817601798623391615488, −8.552582210654185169947910432282, −7.78725762397834399845501950383, −6.97215552905252489550825996785, −5.69357631506729770441793625346, −4.60404267197361776198392826212, −4.09927705974081351998068058053, −2.96356887024572872788567691359, −1.61710585987036550665790188428, 0.937650252359840675215412182336, 2.62031745727679009782329375122, 4.13438303197822308612073670959, 4.70710987721226500240081324973, 5.77634075264069230582302839506, 6.32227318810093452961311734622, 7.58808433244653552947176096999, 8.444666103243149812117003760367, 9.110249063856878812458536096410, 10.47101143342407641612894742496

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