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Properties

Label 2-495-45.4-c1-0-36
Degree $2$
Conductor $495$
Sign $0.970 + 0.239i$
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  + (−0.556 + 0.321i)2-s + (1.67 − 0.452i)3-s + (−0.793 + 1.37i)4-s + (−2.03 − 0.916i)5-s + (−0.784 + 0.789i)6-s + (1.85 − 1.06i)7-s − 2.30i·8-s + (2.58 − 1.51i)9-s + (1.42 − 0.145i)10-s + (−0.5 − 0.866i)11-s + (−0.704 + 2.65i)12-s + (4.17 + 2.40i)13-s + (−0.686 + 1.18i)14-s + (−3.82 − 0.608i)15-s + (−0.846 − 1.46i)16-s − 5.87i·17-s + ⋯
L(s)  = 1  + (−0.393 + 0.227i)2-s + (0.965 − 0.261i)3-s + (−0.396 + 0.687i)4-s + (−0.912 − 0.409i)5-s + (−0.320 + 0.322i)6-s + (0.699 − 0.403i)7-s − 0.814i·8-s + (0.863 − 0.504i)9-s + (0.451 − 0.0459i)10-s + (−0.150 − 0.261i)11-s + (−0.203 + 0.767i)12-s + (1.15 + 0.668i)13-s + (−0.183 + 0.317i)14-s + (−0.987 − 0.157i)15-s + (−0.211 − 0.366i)16-s − 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.970 + 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39510 - 0.169882i\)
\(L(\frac12)\) \(\approx\) \(1.39510 - 0.169882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.452i)T \)
5 \( 1 + (2.03 + 0.916i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.556 - 0.321i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-1.85 + 1.06i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-4.17 - 2.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.87iT - 17T^{2} \)
19 \( 1 + 0.164T + 19T^{2} \)
23 \( 1 + (-7.32 - 4.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.178 - 0.309i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.521 + 0.902i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.83iT - 37T^{2} \)
41 \( 1 + (1.68 - 2.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.67 + 0.965i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.68 - 3.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.75iT - 53T^{2} \)
59 \( 1 + (-4.52 + 7.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.22 - 5.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.51 + 3.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 9.81iT - 73T^{2} \)
79 \( 1 + (-4.17 - 7.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.85 + 1.64i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.21T + 89T^{2} \)
97 \( 1 + (7.26 - 4.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.10842402919234195113316841009, −9.532914161157307001037452073040, −8.904780896396792334286515808639, −8.251444355451491473706360263612, −7.49729460914321361436890140921, −6.90310932375544110765904814934, −4.88933725452569590378869339342, −3.97234804155672557562482543037, −3.11777041598713691226720389759, −1.08226177438968388346851059282, 1.46545361337958177276611759748, 2.93252966373564811848668776284, 4.13334385104858741218269050210, 5.08211322508217251533435775118, 6.44897694025127529041044351788, 7.80196707150433549264239887194, 8.506884371482820366525765225925, 8.887894373135397921443466408854, 10.37939713468368615477670866728, 10.59580554863521042455108264848

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