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Properties

Label 2-495-11.3-c1-0-10
Degree $2$
Conductor $495$
Sign $0.959 - 0.281i$
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  + (−1.90 + 1.38i)2-s + (1.09 − 3.37i)4-s + (0.809 + 0.587i)5-s + (0.0598 − 0.184i)7-s + (1.12 + 3.47i)8-s − 2.35·10-s + (1.96 − 2.67i)11-s + (−0.787 + 0.572i)13-s + (0.140 + 0.433i)14-s + (−1.21 − 0.880i)16-s + (−2.16 − 1.57i)17-s + (−1.71 − 5.27i)19-s + (2.87 − 2.08i)20-s + (−0.0369 + 7.81i)22-s + 4.80·23-s + ⋯
L(s)  = 1  + (−1.34 + 0.979i)2-s + (0.548 − 1.68i)4-s + (0.361 + 0.262i)5-s + (0.0226 − 0.0695i)7-s + (0.398 + 1.22i)8-s − 0.744·10-s + (0.591 − 0.806i)11-s + (−0.218 + 0.158i)13-s + (0.0376 + 0.115i)14-s + (−0.303 − 0.220i)16-s + (−0.525 − 0.381i)17-s + (−0.393 − 1.21i)19-s + (0.642 − 0.466i)20-s + (−0.00787 + 1.66i)22-s + 1.00·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.959 - 0.281i$
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: \((2,\ 495,\ (\ :1/2),\ 0.959 - 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731426 + 0.105183i\)
\(L(\frac12)\) \(\approx\) \(0.731426 + 0.105183i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 + 2.67i)T \)
good2 \( 1 + (1.90 - 1.38i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.0598 + 0.184i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.787 - 0.572i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.16 + 1.57i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.71 + 5.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.80T + 23T^{2} \)
29 \( 1 + (-3.12 + 9.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.02 + 1.47i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.76 + 5.43i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.55 - 7.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 + (-3.35 - 10.3i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.51 - 5.45i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.46 + 10.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.975 + 0.708i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.25T + 67T^{2} \)
71 \( 1 + (-4.84 - 3.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.02 + 3.14i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.21 - 5.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.72 - 4.88i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.34T + 89T^{2} \)
97 \( 1 + (-12.8 + 9.31i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.90348508799922648753875939335, −9.619981404551656276920411416387, −9.242052350927133682516251289486, −8.334486588194931356421453928080, −7.40194186362945841121533420376, −6.54005062527653531242682984781, −5.92468110172466078398576852312, −4.48000453085394229044654644196, −2.62289979416169318041374406289, −0.803694422045447046168057381738, 1.29084104248400969183126189209, 2.35271965290934914839345732267, 3.73276058495910816372920014597, 5.13956879123476270008603337438, 6.60690528789208532729307736469, 7.56530662951594586972472330748, 8.710202315709277871432579012616, 9.059338526386513421013073643215, 10.21670645046908334961097641235, 10.49640217623485267489006612127

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