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Properties

Label 2-1449-161.27-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.523487113\)
\(L(\frac12)\) \(\approx\) \(1.523487113\)
\(L(1)\) 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

−9.670077040461811308013927926258, −9.009062472348048823234581371255, −8.158355746676953407098028025450, −7.56201960681698601211249245566, −6.63942931786595102275311015415, −5.74560835968242505181004130008, −5.47822049941964471406657989783, −4.34307036012119662106779134885, −3.41348371864380664062202379103, −1.97798315189398451564440292380, 1.14186110869076636067575625486, 2.26287029125150532460292332547, 3.31850846987658587647796765012, 4.25686149576791620424215546143, 4.69097840979598112966213372010, 5.92400737460783033258286232676, 7.04884466928912351187108309340, 7.71562680499792986404712912602, 8.908600313508296539871320997301, 9.733408889361485074091121283881

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