↑

Properties

Label 2-2299-209.75-c0-0-1
Degree $2$
Conductor $2299$
Sign $-0.605 + 0.795i$
Analytic cond. $1.14735$
Root an. cond. $1.07114$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)4-s + (0.190 + 0.587i)5-s + (−0.5 − 0.363i)7-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)16-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)19-s + (0.190 − 0.587i)20-s − 1.61·23-s + (0.5 − 0.363i)25-s + (0.190 + 0.587i)28-s + (0.118 − 0.363i)35-s + (−0.809 + 0.587i)36-s − 1.61·43-s + 0.618·45-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)4-s + (0.190 + 0.587i)5-s + (−0.5 − 0.363i)7-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)16-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)19-s + (0.190 − 0.587i)20-s − 1.61·23-s + (0.5 − 0.363i)25-s + (0.190 + 0.587i)28-s + (0.118 − 0.363i)35-s + (−0.809 + 0.587i)36-s − 1.61·43-s + 0.618·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2299\)    =    \(11^{2} \cdot 19\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(1.14735\)
Root analytic conductor: \(1.07114\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2299} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2299,\ (\ :0),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5854531251\)
\(L(\frac12)\) \(\approx\) \(0.5854531251\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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

−9.036434943868197656636424138324, −8.326650845070205468141599623091, −7.19220889424802501825216545624, −6.51166557420578610428987001252, −5.95482929886871154030739677517, −4.85951584936671409535559929012, −4.05494845456407500605766765872, −3.26764356545183373928844597416, −1.95975647380964576750730654769, −0.39863809631863444771520194747, 1.70480570409335260650239137396, 2.80613237758966114862270844949, 4.05922477513710240919244475444, 4.49856852194841064064957032611, 5.48464943269117494472339248293, 6.24135332316311305865840810471, 7.30883406521074805585841241367, 8.192987520605901778606313645743, 8.612223202585879265427825097895, 9.292022525918347265006775555594

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