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Properties

Label 2-1449-483.293-c0-0-0
Degree $2$
Conductor $1449$
Sign $0.955 - 0.294i$
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  + (−1.81 − 0.828i)2-s + (1.95 + 2.25i)4-s + (−0.989 − 0.142i)7-s + (−1.11 − 3.78i)8-s + (0.398 + 0.871i)11-s + (1.67 + 1.07i)14-s + (−0.697 + 4.84i)16-s − 1.91i·22-s + (−0.212 + 0.977i)23-s + (0.415 − 0.909i)25-s + (−1.61 − 2.50i)28-s + (0.528 + 0.457i)29-s + (3.14 − 4.89i)32-s + (0.153 − 0.239i)37-s + (−0.474 + 1.61i)43-s + (−1.18 + 2.59i)44-s + ⋯
L(s)  = 1  + (−1.81 − 0.828i)2-s + (1.95 + 2.25i)4-s + (−0.989 − 0.142i)7-s + (−1.11 − 3.78i)8-s + (0.398 + 0.871i)11-s + (1.67 + 1.07i)14-s + (−0.697 + 4.84i)16-s − 1.91i·22-s + (−0.212 + 0.977i)23-s + (0.415 − 0.909i)25-s + (−1.61 − 2.50i)28-s + (0.528 + 0.457i)29-s + (3.14 − 4.89i)32-s + (0.153 − 0.239i)37-s + (−0.474 + 1.61i)43-s + (−1.18 + 2.59i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.955 - 0.294i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ 0.955 - 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3846793620\)
\(L(\frac12)\) \(\approx\) \(0.3846793620\)
\(L(1)\) 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 \( 1 + (0.989 + 0.142i)T \)
23 \( 1 + (0.212 - 0.977i)T \)
good2 \( 1 + (1.81 + 0.828i)T + (0.654 + 0.755i)T^{2} \)
5 \( 1 + (-0.415 + 0.909i)T^{2} \)
11 \( 1 + (-0.398 - 0.871i)T + (-0.654 + 0.755i)T^{2} \)
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 \( 1 + (-0.528 - 0.457i)T + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.153 + 0.239i)T + (-0.415 - 0.909i)T^{2} \)
41 \( 1 + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (0.474 - 1.61i)T + (-0.841 - 0.540i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.266 - 1.85i)T + (-0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (-1.37 - 0.627i)T + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (-1.59 - 0.729i)T + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (1.80 - 0.258i)T + (0.959 - 0.281i)T^{2} \)
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

−9.799291434792522329312019392520, −9.178823568573162714730426049981, −8.393356685512226629084743284287, −7.51283838015695647963126248661, −6.87933317163169653161225820960, −6.16175482564136190097956968989, −4.26642149246153208998834722608, −3.28757933577220266076495775328, −2.42090044317300400512978788630, −1.19437157157134077739254055765, 0.63908677029181319737673201440, 2.13846457942814704375996858182, 3.31470258673000310667340213396, 5.15215502232577000272433109694, 6.06266945307195745116397847022, 6.60456622349636286475260036800, 7.27657808499511459278586383550, 8.385876633731001325403638057474, 8.704662887135624881012799950061, 9.596733632301856764344215002996

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