↑

Properties

Label 2-819-91.4-c1-0-32
Degree $2$
Conductor $819$
Sign $0.993 - 0.110i$
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  + 0.411i·2-s + 1.83·4-s + (1.54 − 0.892i)5-s + (2.44 + 1.00i)7-s + 1.57i·8-s + (0.367 + 0.636i)10-s + (4.35 − 2.51i)11-s + (−0.0961 − 3.60i)13-s + (−0.412 + 1.00i)14-s + 3.01·16-s − 7.92·17-s + (2.74 + 1.58i)19-s + (2.82 − 1.63i)20-s + (1.03 + 1.79i)22-s − 7.89·23-s + ⋯
L(s)  = 1  + 0.291i·2-s + 0.915·4-s + (0.691 − 0.398i)5-s + (0.925 + 0.379i)7-s + 0.557i·8-s + (0.116 + 0.201i)10-s + (1.31 − 0.758i)11-s + (−0.0266 − 0.999i)13-s + (−0.110 + 0.269i)14-s + 0.752·16-s − 1.92·17-s + (0.628 + 0.362i)19-s + (0.632 − 0.365i)20-s + (0.220 + 0.382i)22-s − 1.64·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41724 + 0.134238i\)
\(L(\frac12)\) \(\approx\) \(2.41724 + 0.134238i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0961 + 3.60i)T \)
good2 \( 1 - 0.411iT - 2T^{2} \)
5 \( 1 + (-1.54 + 0.892i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.35 + 2.51i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.92T + 17T^{2} \)
19 \( 1 + (-2.74 - 1.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.89T + 23T^{2} \)
29 \( 1 + (-2.23 + 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.24 + 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.26iT - 37T^{2} \)
41 \( 1 + (2.53 + 1.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.50 + 2.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.196 - 0.113i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.51 - 4.35i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.21iT - 59T^{2} \)
61 \( 1 + (0.128 - 0.222i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.55 + 3.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.64 + 2.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.59 + 2.65i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.43 - 9.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 - 5.98iT - 89T^{2} \)
97 \( 1 + (3.99 - 2.30i)T + (48.5 - 84.0i)T^{2} \)
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

−10.29657161906276404438751195587, −9.300677386522207714074043008792, −8.439845460697028963487337050959, −7.81073381314474633247879991999, −6.56653349114381851209285825533, −5.96953196178978035226383971349, −5.16802155254458574880338735843, −3.85652747395558300763247419339, −2.38897882393573099194406707034, −1.46971379268872760020434732499, 1.73613503433380271516475900111, 2.14305854058860699493351154490, 3.82393545393995618439611192249, 4.66316870576371656506509046251, 6.11957724415997575344502396140, 6.78188313513885082176008914536, 7.35211751941358748981955169703, 8.668609415786979102451564864566, 9.515701707557853655093775040243, 10.31016944551896258889997135502

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