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Properties

Label 2-495-11.4-c1-0-10
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: $\chi_{495} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02698 + 2.27395i\)
\(L(\frac12)\) \(\approx\) \(2.02698 + 2.27395i\)
\(L(\frac{3}{2})\) 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

−11.70024107188218320480139883432, −10.20330629580559540991498481400, −9.373362397694523511875539904135, −7.978733627777698467078078666912, −7.47777069430928805314273638216, −6.32832727705378533673322144403, −5.24649365326114507955452655360, −5.06692583734787115159116451211, −3.58879068171545203473623342825, −2.33758058736319452370348987734, 1.46822349482902458714478397122, 2.85641962355723849111358498443, 3.71619983718808162204403972127, 4.87820666957428580623924814867, 5.63651227957942136939447591353, 6.72560602292743987367887590424, 7.87262923825626921304124440892, 9.240675652923897305926578722010, 10.42553716485776400992138771630, 10.75460725284387752101176211542

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