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Properties

Label 2-1449-161.118-c0-0-2
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

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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

\(L(\frac{1}{2})\) \(\approx\) \(1.118432545\)
\(L(\frac12)\) \(\approx\) \(1.118432545\)
\(L(1)\) 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 \( 1 + (0.142 + 0.989i)T^{2} \)
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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

−9.707794056129962139340299941526, −8.713088252918194837555444804836, −8.173409461332418674822630172105, −6.77627818212203934131610353234, −6.26732807175153591244852095535, −5.66110365614696100858024765923, −4.17137237905461735568808948894, −3.07805157725388720519155709052, −2.59066254499092634632145626325, −0.957991826559447408630633955205, 1.78863146477091518100172262757, 2.89031107402610917373171494034, 4.00268551998330568734956010405, 4.96714423577553962329009362599, 6.20484472212194698842792784547, 6.63664595727048511919598466198, 7.55765158606998476662561458014, 7.891003526474704344107293553142, 9.315190594101220197938276782411, 9.732503430153216673710511300630

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