↑

Properties

Label 2-1449-161.118-c0-0-0
Degree $2$
Conductor $1449$
Sign $0.0367 - 0.999i$
Analytic cond. $0.723145$
Root an. cond. $0.850379$
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  + (0.0801 + 0.557i)2-s + (0.654 − 0.192i)4-s + (−0.841 + 0.540i)7-s + (0.393 + 0.862i)8-s + (−0.258 + 1.80i)11-s + (−0.368 − 0.425i)14-s + (0.124 − 0.0801i)16-s − 1.02·22-s + (0.755 + 0.654i)23-s + (−0.142 − 0.989i)25-s + (−0.446 + 0.515i)28-s + (−1.89 − 0.557i)29-s + (0.675 + 0.779i)32-s + (1.10 + 1.27i)37-s + (0.544 − 1.19i)43-s + (0.176 + 1.22i)44-s + ⋯
L(s)  = 1  + (0.0801 + 0.557i)2-s + (0.654 − 0.192i)4-s + (−0.841 + 0.540i)7-s + (0.393 + 0.862i)8-s + (−0.258 + 1.80i)11-s + (−0.368 − 0.425i)14-s + (0.124 − 0.0801i)16-s − 1.02·22-s + (0.755 + 0.654i)23-s + (−0.142 − 0.989i)25-s + (−0.446 + 0.515i)28-s + (−1.89 − 0.557i)29-s + (0.675 + 0.779i)32-s + (1.10 + 1.27i)37-s + (0.544 − 1.19i)43-s + (0.176 + 1.22i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0367 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0367 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.0367 - 0.999i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ 0.0367 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229292540\)
\(L(\frac12)\) \(\approx\) \(1.229292540\)
\(L(1)\) 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 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.755 - 0.654i)T \)
good2 \( 1 + (-0.0801 - 0.557i)T + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (0.258 - 1.80i)T + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (1.89 + 0.557i)T + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 - 0.755i)T^{2} \)
37 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (-0.544 + 1.19i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.66 + 1.07i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.142 + 0.989i)T^{2} \)
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

−9.869181750347702679158870320716, −9.215517462775490446683921463119, −8.095317001324338779166469333530, −7.27051791741832597718804077211, −6.79943282137967673154772446956, −5.85202216752280500062603203322, −5.17229808064660226088160214618, −4.06839379918072014537114648397, −2.69841619763634828568108896735, −1.92658310284431738106059598030, 0.995654484774723581104515532422, 2.57294303339651321958543387570, 3.36074711741610892155578071188, 4.00872352288323775806993857238, 5.56770699366694530821983940128, 6.21241561705609033378601636003, 7.15730567534287618209877076217, 7.75959683171630554812028371438, 8.909184646693022331535142139754, 9.555738612442769085345061027773

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