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Properties

Label 2-1449-161.118-c0-0-1
Degree $2$
Conductor $1449$
Sign $-0.999 - 0.0367i$
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  + (−0.273 − 1.89i)2-s + (−2.57 + 0.755i)4-s + (0.841 − 0.540i)7-s + (1.34 + 2.93i)8-s + (0.118 − 0.822i)11-s + (−1.25 − 1.45i)14-s + (2.95 − 1.89i)16-s − 1.59·22-s + (0.654 − 0.755i)23-s + (−0.142 − 0.989i)25-s + (−1.75 + 2.02i)28-s + (−0.273 − 0.0801i)29-s + (−2.30 − 2.65i)32-s + (−1.10 − 1.27i)37-s + (−0.544 + 1.19i)43-s + (0.317 + 2.20i)44-s + ⋯
L(s)  = 1  + (−0.273 − 1.89i)2-s + (−2.57 + 0.755i)4-s + (0.841 − 0.540i)7-s + (1.34 + 2.93i)8-s + (0.118 − 0.822i)11-s + (−1.25 − 1.45i)14-s + (2.95 − 1.89i)16-s − 1.59·22-s + (0.654 − 0.755i)23-s + (−0.142 − 0.989i)25-s + (−1.75 + 2.02i)28-s + (−0.273 − 0.0801i)29-s + (−2.30 − 2.65i)32-s + (−1.10 − 1.27i)37-s + (−0.544 + 1.19i)43-s + (0.317 + 2.20i)44-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $-0.999 - 0.0367i$
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: \((2,\ 1449,\ (\ :0),\ -0.999 - 0.0367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8319771033\)
\(L(\frac12)\) \(\approx\) \(0.8319771033\)
\(L(1)\) 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 \( 1 + (-0.654 + 0.755i)T \)
good2 \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
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 \( 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2} \)
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 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
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 \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \)
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 \( 1 + (0.142 + 0.989i)T^{2} \)
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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

−9.556418498211508410082553340470, −8.561799367683249756041482159472, −8.306035348024326309118324927677, −7.13073244259021093231125756615, −5.64012209518999218208183733724, −4.66143516245392365601576160675, −3.97784613049522252427781317306, −3.01726960438066329465561117161, −1.98461432084341511118239381917, −0.836929612933149708451958860686, 1.62469467979354561567526027413, 3.63019852718823939762924433609, 4.86515348100597471167695017579, 5.17278065527599371257089806533, 6.13148621177132994081419966509, 7.05784296502839814302655146172, 7.56058319013262546604540048524, 8.383923423514885587826516962025, 9.041989751472825156254429631263, 9.658533569259674354938703931972

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