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Properties

Label 2-495-11.3-c1-0-9
Degree $2$
Conductor $495$
Sign $0.908 - 0.418i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.991 − 0.720i)2-s + (−0.153 + 0.472i)4-s + (0.809 + 0.587i)5-s + (−0.139 + 0.429i)7-s + (0.945 + 2.91i)8-s + 1.22·10-s + (−1.55 + 2.93i)11-s + (3.91 − 2.84i)13-s + (0.171 + 0.526i)14-s + (2.23 + 1.62i)16-s + (−0.598 − 0.435i)17-s + (2.10 + 6.47i)19-s + (−0.402 + 0.292i)20-s + (0.572 + 4.02i)22-s − 0.00634·23-s + ⋯
L(s)  = 1  + (0.701 − 0.509i)2-s + (−0.0767 + 0.236i)4-s + (0.361 + 0.262i)5-s + (−0.0527 + 0.162i)7-s + (0.334 + 1.02i)8-s + 0.387·10-s + (−0.468 + 0.883i)11-s + (1.08 − 0.789i)13-s + (0.0457 + 0.140i)14-s + (0.558 + 0.405i)16-s + (−0.145 − 0.105i)17-s + (0.482 + 1.48i)19-s + (−0.0898 + 0.0653i)20-s + (0.122 + 0.858i)22-s − 0.00132·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.908 - 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03279 + 0.445641i\)
\(L(\frac12)\) \(\approx\) \(2.03279 + 0.445641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (1.55 - 2.93i)T \)
good2 \( 1 + (-0.991 + 0.720i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.139 - 0.429i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.91 + 2.84i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.598 + 0.435i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.10 - 6.47i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.00634T + 23T^{2} \)
29 \( 1 + (0.100 - 0.308i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.53 + 3.29i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.27 + 7.00i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.39 + 10.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.80T + 43T^{2} \)
47 \( 1 + (0.518 + 1.59i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.98 - 5.07i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.463 - 1.42i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.5 + 7.66i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.60T + 67T^{2} \)
71 \( 1 + (9.23 + 6.70i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.16 + 9.73i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.69 + 1.23i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.8 - 9.34i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.36T + 89T^{2} \)
97 \( 1 + (4.75 - 3.45i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.98952822711531797193700976835, −10.40346230731241794279804812701, −9.337755632481345643066838698483, −8.197202821677847827359712631968, −7.51470363447875901178755418798, −6.07740202165578278922821569396, −5.27759704485057092140121761286, −4.07785170019308857095877258743, −3.12372392842868037598141737382, −1.93343746196572892769213435764, 1.15161825299463361527328229204, 3.10300840809347635827001295514, 4.39854465889474649593397153173, 5.21555702062260132464667682434, 6.24716761443637348673965312481, 6.80321971271261960133808014321, 8.200054508244830469330520020151, 9.083528768108784844569077408150, 9.974839429337604130172217814775, 10.94058076391149749264300738791

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