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Properties

Label 2-1449-483.251-c0-0-1
Degree $2$
Conductor $1449$
Sign $0.0925 - 0.995i$
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  + (−0.377 + 0.587i)2-s + (0.212 + 0.465i)4-s + (0.755 + 0.654i)7-s + (−1.04 − 0.150i)8-s + (1.34 − 0.865i)11-s + (−0.670 + 0.196i)14-s + (0.148 − 0.171i)16-s + 1.11i·22-s + (0.479 + 0.877i)23-s + (0.841 + 0.540i)25-s + (−0.144 + 0.490i)28-s + (−0.386 − 0.176i)29-s + (−0.252 − 0.861i)32-s + (−0.368 − 1.25i)37-s + (−1.89 + 0.273i)43-s + (0.688 + 0.442i)44-s + ⋯
L(s)  = 1  + (−0.377 + 0.587i)2-s + (0.212 + 0.465i)4-s + (0.755 + 0.654i)7-s + (−1.04 − 0.150i)8-s + (1.34 − 0.865i)11-s + (−0.670 + 0.196i)14-s + (0.148 − 0.171i)16-s + 1.11i·22-s + (0.479 + 0.877i)23-s + (0.841 + 0.540i)25-s + (−0.144 + 0.490i)28-s + (−0.386 − 0.176i)29-s + (−0.252 − 0.861i)32-s + (−0.368 − 1.25i)37-s + (−1.89 + 0.273i)43-s + (0.688 + 0.442i)44-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.0925 - 0.995i$
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: \((2,\ 1449,\ (\ :0),\ 0.0925 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068866338\)
\(L(\frac12)\) \(\approx\) \(1.068866338\)
\(L(1)\) 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 \( 1 + (-0.479 - 0.877i)T \)
good2 \( 1 + (0.377 - 0.587i)T + (-0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (-1.34 + 0.865i)T + (0.415 - 0.909i)T^{2} \)
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 \( 1 + (0.386 + 0.176i)T + (0.654 + 0.755i)T^{2} \)
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 \( 1 + (1.27 - 1.47i)T + (-0.142 - 0.989i)T^{2} \)
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 \( 1 + (0.647 - 1.00i)T + (-0.415 - 0.909i)T^{2} \)
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

−9.394596291750660523558805108390, −9.033979485556560716172378254211, −8.321765863191474809523989436031, −7.56596874960887830930309113590, −6.71770197136489272143289190191, −5.98657131290264730397279015084, −5.10767737035074514265996343929, −3.82001590584686660969509029503, −2.99746944988031989932903704423, −1.57168449837850270584372571007, 1.17668591613651340903955471015, 2.01382103178319841296162832970, 3.34382984345883925878396786448, 4.48025205980811859378752586036, 5.19460477138020047181464784658, 6.70449717878916650692256442722, 6.73559026509797794251278146200, 8.148325642099294095173239527478, 8.852489190048977798986421230143, 9.715186742978799610593381424388

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