↑

Properties

Label 2-819-7.4-c1-0-21
Degree $2$
Conductor $819$
Sign $-0.605 - 0.795i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 2.26i)2-s + (−2.42 + 4.20i)4-s + (1.11 + 1.93i)5-s + (−2 + 1.73i)7-s + 7.47·8-s + (2.92 − 5.06i)10-s + (−1.5 + 2.59i)11-s − 13-s + (6.54 + 2.26i)14-s + (−4.92 − 8.53i)16-s + (0.736 − 1.27i)17-s + (−1.5 − 2.59i)19-s − 10.8·20-s + 7.85·22-s + (−4.11 − 7.13i)23-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)2-s + (−1.21 + 2.10i)4-s + (0.499 + 0.866i)5-s + (−0.755 + 0.654i)7-s + 2.64·8-s + (0.925 − 1.60i)10-s + (−0.452 + 0.783i)11-s − 0.277·13-s + (1.74 + 0.605i)14-s + (−1.23 − 2.13i)16-s + (0.178 − 0.309i)17-s + (−0.344 − 0.596i)19-s − 2.42·20-s + 1.67·22-s + (−0.858 − 1.48i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.30 + 2.26i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.736 + 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.11 + 7.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.35 + 4.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (3.73 + 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.73 - 6.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.736 - 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + (-1.35 + 2.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.35 - 2.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.11 - 1.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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

−9.876489472587370965440491531903, −9.213795769930802814763199912084, −8.379930000021349043469553804432, −7.29600919162277366229111860774, −6.38106465960017753553064942877, −4.92359572477605806208166223835, −3.61803831127645157847767493753, −2.60344181598290445483504774027, −2.10763991469750759528048455278, 0, 1.44753318114864922335316415897, 3.72307114132072439895375022139, 5.07413648051854586608028194205, 5.78329784947620868191492817450, 6.44397900283482383999943819474, 7.55980208973122466636735299024, 8.071750805877635202654697591420, 9.022596016919218622266460582322, 9.685189091288742995533559344134

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