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Properties

Label 2-1449-483.251-c0-0-3
Degree $2$
Conductor $1449$
Sign $-0.995 - 0.0925i$
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  + (1.01 − 1.57i)2-s + (−1.04 − 2.28i)4-s + (−0.755 − 0.654i)7-s + (−2.80 − 0.403i)8-s + (1.00 − 0.647i)11-s + (−1.79 + 0.527i)14-s + (−1.83 + 2.11i)16-s − 2.24i·22-s + (−0.877 + 0.479i)23-s + (0.841 + 0.540i)25-s + (−0.707 + 2.40i)28-s + (−1.77 − 0.811i)29-s + (0.678 + 2.30i)32-s + (0.368 + 1.25i)37-s + (1.89 − 0.273i)43-s + (−2.53 − 1.62i)44-s + ⋯
L(s)  = 1  + (1.01 − 1.57i)2-s + (−1.04 − 2.28i)4-s + (−0.755 − 0.654i)7-s + (−2.80 − 0.403i)8-s + (1.00 − 0.647i)11-s + (−1.79 + 0.527i)14-s + (−1.83 + 2.11i)16-s − 2.24i·22-s + (−0.877 + 0.479i)23-s + (0.841 + 0.540i)25-s + (−0.707 + 2.40i)28-s + (−1.77 − 0.811i)29-s + (0.678 + 2.30i)32-s + (0.368 + 1.25i)37-s + (1.89 − 0.273i)43-s + (−2.53 − 1.62i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $-0.995 - 0.0925i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ -0.995 - 0.0925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530706749\)
\(L(\frac12)\) \(\approx\) \(1.530706749\)
\(L(1)\) 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 \( 1 + (0.755 + 0.654i)T \)
23 \( 1 + (0.877 - 0.479i)T \)
good2 \( 1 + (-1.01 + 1.57i)T + (-0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (-1.00 + 0.647i)T + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (1.77 + 0.811i)T + (0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.959 + 0.281i)T^{2} \)
37 \( 1 + (-0.368 - 1.25i)T + (-0.841 + 0.540i)T^{2} \)
41 \( 1 + (0.841 + 0.540i)T^{2} \)
43 \( 1 + (-1.89 + 0.273i)T + (0.959 - 0.281i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.278 + 0.321i)T + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.983 + 1.53i)T + (-0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.865 + 1.34i)T + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.959 - 0.281i)T^{2} \)
97 \( 1 + (0.841 + 0.540i)T^{2} \)
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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

−9.492692735559927289218594702099, −9.103681438289656662551190710554, −7.66571125313418177616841972629, −6.43243346986035691908353219341, −5.86098925720350066042999326600, −4.77487747551135621730617703884, −3.75457492054396163119941894908, −3.47642641362681037695580516665, −2.19281062979212818449668970036, −0.952610677677601539265423321860, 2.48428074261349951390717852667, 3.73490850982440766889634270279, 4.29282036540244441420270291837, 5.45143461054317506137404328653, 5.99227199298409301490586366718, 6.82902429489693055807379977314, 7.32149313779318373537031996128, 8.376311944748307201109950555497, 9.080984679643572705844240350350, 9.682536017163582117566592897432

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