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Properties

Label 2-1449-161.146-c0-0-1
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: $\chi_{1449} (307, \cdot )$
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

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

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