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Properties

Label 2-495-11.3-c1-0-13
Degree $2$
Conductor $495$
Sign $-0.114 + 0.993i$
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  + (−1.93 + 1.40i)2-s + (1.15 − 3.55i)4-s + (−0.809 − 0.587i)5-s + (0.608 − 1.87i)7-s + (1.28 + 3.95i)8-s + 2.39·10-s + (1.69 + 2.84i)11-s + (−2.36 + 1.71i)13-s + (1.45 + 4.48i)14-s + (−2.00 − 1.45i)16-s + (−5.35 − 3.89i)17-s + (−1.10 − 3.38i)19-s + (−3.02 + 2.19i)20-s + (−7.30 − 3.13i)22-s − 2.78·23-s + ⋯
L(s)  = 1  + (−1.37 + 0.995i)2-s + (0.577 − 1.77i)4-s + (−0.361 − 0.262i)5-s + (0.230 − 0.708i)7-s + (0.454 + 1.39i)8-s + 0.757·10-s + (0.511 + 0.859i)11-s + (−0.655 + 0.476i)13-s + (0.389 + 1.19i)14-s + (−0.502 − 0.364i)16-s + (−1.29 − 0.943i)17-s + (−0.252 − 0.776i)19-s + (−0.675 + 0.490i)20-s + (−1.55 − 0.667i)22-s − 0.581·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 495,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149069 - 0.167232i\)
\(L(\frac12)\) \(\approx\) \(0.149069 - 0.167232i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 2.84i)T \)
good2 \( 1 + (1.93 - 1.40i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.608 + 1.87i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.36 - 1.71i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.35 + 3.89i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.10 + 3.38i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.78T + 23T^{2} \)
29 \( 1 + (-1.08 + 3.34i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.56 - 4.77i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.36 + 4.21i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.18 + 6.73i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (-0.00839 - 0.0258i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.32 + 2.41i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.94 + 9.04i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.50 + 1.82i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 1.58T + 67T^{2} \)
71 \( 1 + (4.90 + 3.56i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.733 - 2.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.1 + 7.37i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.08 + 2.96i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (8.42 - 6.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.42824811123230105008073281700, −9.468757419785183797716738115983, −8.973426019384813239566919747235, −7.952002772367330943866765463076, −6.99362779519268519929425461453, −6.80306868606327730755571079377, −5.14922874424323105140981454199, −4.16827962659178074221562212320, −1.93439612979499160636113882985, −0.20481243555999796316986383133, 1.71147405060159010456523058955, 2.85535301101758411832376947373, 4.00597701819109917378777581848, 5.70407909617648610270588234654, 6.90265423600630726541933503339, 8.190078035188119592565224128848, 8.472599984307006338710924116031, 9.457823353458133081272092543699, 10.34255476414923208334627229521, 11.07750324202393072083502354023

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