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Properties

Label 2-1449-161.27-c0-0-0
Degree $2$
Conductor $1449$
Sign $0.739 + 0.673i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.627 − 1.37i)2-s + (−0.841 + 0.970i)4-s + (0.142 + 0.989i)7-s + (0.412 + 0.121i)8-s + (−0.234 + 0.512i)11-s + (1.27 − 0.817i)14-s + (0.0902 + 0.627i)16-s + 0.851·22-s + (0.540 − 0.841i)23-s + (0.415 + 0.909i)25-s + (−1.08 − 0.694i)28-s + (1.19 + 1.37i)29-s + (1.16 − 0.750i)32-s + (0.239 − 0.153i)37-s + (1.61 − 0.474i)43-s + (−0.300 − 0.658i)44-s + ⋯
L(s)  = 1  + (−0.627 − 1.37i)2-s + (−0.841 + 0.970i)4-s + (0.142 + 0.989i)7-s + (0.412 + 0.121i)8-s + (−0.234 + 0.512i)11-s + (1.27 − 0.817i)14-s + (0.0902 + 0.627i)16-s + 0.851·22-s + (0.540 − 0.841i)23-s + (0.415 + 0.909i)25-s + (−1.08 − 0.694i)28-s + (1.19 + 1.37i)29-s + (1.16 − 0.750i)32-s + (0.239 − 0.153i)37-s + (1.61 − 0.474i)43-s + (−0.300 − 0.658i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (1315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ 0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7359115250\)
\(L(\frac12)\) \(\approx\) \(0.7359115250\)
\(L(1)\) 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 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-0.540 + 0.841i)T \)
good2 \( 1 + (0.627 + 1.37i)T + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (-0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.234 - 0.512i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (-1.19 - 1.37i)T + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 - 0.540i)T^{2} \)
37 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.258 - 1.80i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.234 + 0.512i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 - 0.909i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.584773762145159502416898964462, −9.022030521809970608514657079545, −8.478093211701244656701626476240, −7.42122427767367580701525098374, −6.35991476063233801817275021653, −5.31253896144795694758158536515, −4.36342553176396867065253370141, −3.08290599074314766186631420113, −2.46681852523998879807757225441, −1.32617732208171907328028113606, 0.854358194135056704154117931930, 2.79387130014774882734922076709, 4.10463745404539321580596232770, 5.04247972270418422076789252351, 6.00020106614727300538214384032, 6.67884017576918034017894608033, 7.46570441350875740497605484539, 8.071090621526140315861547672136, 8.724558985037577359807624759386, 9.679914380871692660881044637080

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