↑

Properties

Label 2-820-820.399-c1-0-80
Degree $2$
Conductor $820$
Sign $0.471 + 0.882i$
Analytic cond. $6.54773$
Root an. cond. $2.55885$
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  + (−1.39 + 0.221i)2-s + (−0.568 + 1.37i)3-s + (1.90 − 0.618i)4-s + (1.01 − 1.99i)5-s + (0.490 − 2.04i)6-s + (−0.632 − 2.63i)7-s + (−2.52 + 1.28i)8-s + (0.559 + 0.559i)9-s + (−0.977 + 3.00i)10-s + (−0.233 + 2.96i)12-s + (1.46 + 3.54i)14-s + (2.15 + 2.52i)15-s + (3.23 − 2.35i)16-s + (−0.905 − 0.657i)18-s + (0.699 − 4.41i)20-s + (3.97 + 0.629i)21-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)2-s + (−0.328 + 0.792i)3-s + (0.951 − 0.309i)4-s + (0.453 − 0.891i)5-s + (0.200 − 0.834i)6-s + (−0.239 − 0.995i)7-s + (−0.891 + 0.453i)8-s + (0.186 + 0.186i)9-s + (−0.309 + 0.951i)10-s + (−0.0673 + 0.855i)12-s + (0.391 + 0.946i)14-s + (0.557 + 0.652i)15-s + (0.809 − 0.587i)16-s + (−0.213 − 0.155i)18-s + (0.156 − 0.987i)20-s + (0.867 + 0.137i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.471 + 0.882i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.471 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678706 - 0.406899i\)
\(L(\frac12)\) \(\approx\) \(0.678706 - 0.406899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.221i)T \)
5 \( 1 + (-1.01 + 1.99i)T \)
41 \( 1 + (5.57 + 3.14i)T \)
good3 \( 1 + (0.568 - 1.37i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.632 + 2.63i)T + (-6.23 + 3.17i)T^{2} \)
11 \( 1 + (1.72 + 10.8i)T^{2} \)
13 \( 1 + (-5.90 + 11.5i)T^{2} \)
17 \( 1 + (-16.7 + 2.65i)T^{2} \)
19 \( 1 + (-8.62 - 16.9i)T^{2} \)
23 \( 1 + (-4.99 + 6.88i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (9.35 + 0.736i)T + (28.6 + 4.53i)T^{2} \)
31 \( 1 + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
43 \( 1 + (-8.49 + 1.34i)T + (40.8 - 13.2i)T^{2} \)
47 \( 1 + (-1.73 + 7.20i)T + (-41.8 - 21.3i)T^{2} \)
53 \( 1 + (52.3 + 8.29i)T^{2} \)
59 \( 1 + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.24 + 14.1i)T + (-58.0 - 18.8i)T^{2} \)
67 \( 1 + (-4.15 + 3.55i)T + (10.4 - 66.1i)T^{2} \)
71 \( 1 + (-11.1 - 70.1i)T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (-2.82 - 11.7i)T + (-79.2 + 40.4i)T^{2} \)
97 \( 1 + (15.1 - 95.8i)T^{2} \)
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

−10.03991132403455906637218532957, −9.354148520452415023748558384485, −8.621022281201924205327703112415, −7.58981731495576449976863789611, −6.78365386877704067994010092688, −5.65876478234709026440850717103, −4.81816196892899248650345144364, −3.74320375245639869271438320131, −2.02110960089713890730203174619, −0.57795562493801103596185765047, 1.44436916070118322786094274457, 2.47566782787601916786149933508, 3.49988638811732034524861773965, 5.65767019906836261237552812935, 6.14662968597233786707893787250, 7.19842465119840878234131536914, 7.52075749507100205859003456128, 8.928929277369502369820672092842, 9.444607840993393595927000216988, 10.27683130846621319571146380191

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