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Properties

Label 2-495-11.3-c1-0-18
Degree $2$
Conductor $495$
Sign $-0.132 + 0.991i$
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  + (2.04 − 1.48i)2-s + (1.35 − 4.15i)4-s + (0.809 + 0.587i)5-s + (0.646 − 1.99i)7-s + (−1.85 − 5.69i)8-s + 2.52·10-s + (1.64 + 2.87i)11-s + (−1.04 + 0.757i)13-s + (−1.63 − 5.02i)14-s + (−5.16 − 3.74i)16-s + (−2.41 − 1.75i)17-s + (0.664 + 2.04i)19-s + (3.53 − 2.57i)20-s + (7.63 + 3.43i)22-s − 8.77·23-s + ⋯
L(s)  = 1  + (1.44 − 1.04i)2-s + (0.675 − 2.07i)4-s + (0.361 + 0.262i)5-s + (0.244 − 0.752i)7-s + (−0.654 − 2.01i)8-s + 0.798·10-s + (0.496 + 0.867i)11-s + (−0.289 + 0.210i)13-s + (−0.436 − 1.34i)14-s + (−1.29 − 0.937i)16-s + (−0.586 − 0.426i)17-s + (0.152 + 0.469i)19-s + (0.791 − 0.574i)20-s + (1.62 + 0.732i)22-s − 1.83·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.132 + 0.991i$
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: \((2,\ 495,\ (\ :1/2),\ -0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12133 - 2.42271i\)
\(L(\frac12)\) \(\approx\) \(2.12133 - 2.42271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 2.87i)T \)
good2 \( 1 + (-2.04 + 1.48i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.646 + 1.99i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.04 - 0.757i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.41 + 1.75i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.664 - 2.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.77T + 23T^{2} \)
29 \( 1 + (-0.189 + 0.582i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.94 + 2.14i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.578 - 1.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.57 - 4.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 + (-2.25 - 6.94i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.38 - 1.72i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.00 + 6.17i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.406 - 0.295i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.80T + 67T^{2} \)
71 \( 1 + (9.14 + 6.64i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.43 - 10.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.33 - 3.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.77 - 6.37i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.32T + 89T^{2} \)
97 \( 1 + (-0.284 + 0.206i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.89716338354318691682279544288, −10.10623818582571468559640744627, −9.493338854082436573093796813785, −7.77395994435933079558901390634, −6.66132871958880119241137970251, −5.80302539592572669617323876607, −4.53100671009751216785595535850, −4.05904487407791995470187751280, −2.64916746568538197057958707652, −1.58366093775201601225502476770, 2.36952804555738933025342172624, 3.69294958205671947613563480489, 4.69418115073864446081323110932, 5.73729358369997874071850514527, 6.15015264721593799004231856889, 7.28513260505348403934089490243, 8.333576061906870680998833637399, 9.001078336268335573407125032546, 10.43577322657034987081337664949, 11.75039611460540468033868826385

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