↑

Properties

Label 2-806-403.211-c1-0-1
Degree $2$
Conductor $806$
Sign $0.236 + 0.971i$
Analytic cond. $6.43594$
Root an. cond. $2.53691$
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  + (0.5 + 0.866i)2-s + (−1.37 + 2.38i)3-s + (−0.499 + 0.866i)4-s + (−0.414 + 0.717i)5-s − 2.75·6-s + 0.270·7-s − 0.999·8-s + (−2.29 − 3.97i)9-s − 0.828·10-s − 3.01·11-s + (−1.37 − 2.38i)12-s + (0.172 − 3.60i)13-s + (0.135 + 0.234i)14-s + (−1.14 − 1.97i)15-s + (−0.5 − 0.866i)16-s − 0.811·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.795 + 1.37i)3-s + (−0.249 + 0.433i)4-s + (−0.185 + 0.320i)5-s − 1.12·6-s + 0.102·7-s − 0.353·8-s + (−0.764 − 1.32i)9-s − 0.262·10-s − 0.907·11-s + (−0.397 − 0.688i)12-s + (0.0479 − 0.998i)13-s + (0.0362 + 0.0627i)14-s + (−0.294 − 0.510i)15-s + (−0.125 − 0.216i)16-s − 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(806\)    =    \(2 \cdot 13 \cdot 31\)
Sign: $0.236 + 0.971i$
Analytic conductor: \(6.43594\)
Root analytic conductor: \(2.53691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{806} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 806,\ (\ :1/2),\ 0.236 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.168839 - 0.132676i\)
\(L(\frac12)\) \(\approx\) \(0.168839 - 0.132676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.172 + 3.60i)T \)
31 \( 1 + (5.16 + 2.07i)T \)
good3 \( 1 + (1.37 - 2.38i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.414 - 0.717i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.270T + 7T^{2} \)
11 \( 1 + 3.01T + 11T^{2} \)
17 \( 1 + 0.811T + 17T^{2} \)
19 \( 1 + 4.74T + 19T^{2} \)
23 \( 1 + (1.25 + 2.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 3.57i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-0.876 + 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + 1.19T + 43T^{2} \)
47 \( 1 + 8.17T + 47T^{2} \)
53 \( 1 + (3.58 - 6.20i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.81T + 59T^{2} \)
61 \( 1 + (-0.754 + 1.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 7.46T + 67T^{2} \)
71 \( 1 + (-0.455 - 0.788i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.29 - 5.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.10 + 8.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.23 - 3.86i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.16 + 5.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.82 - 6.61i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.80655522593184438016058488586, −10.28855030394342192221656805678, −9.304797000099180283528104692138, −8.354272769333982557932808267126, −7.44508441130893113603969312358, −6.28602846066001715212090185817, −5.50855172835442842022770826187, −4.80675857632580620750411796933, −3.93063066062576213835520614956, −2.86745710671007304870830802778, 0.10226094725624830020660759905, 1.57317730347553994434221280576, 2.51839769483293394234745713449, 4.18984737311224716549487364174, 5.12428972264481926823566950076, 6.11051506201607028809446095380, 6.79381835284101067154905914196, 7.82314393024780869370251209537, 8.589980246549618719844123692530, 9.764350669377866689301163151878

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