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Properties

Label 2-495-11.5-c1-0-17
Degree $2$
Conductor $495$
Sign $-0.667 + 0.744i$
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  + (−0.338 − 1.04i)2-s + (0.647 − 0.470i)4-s + (−0.309 + 0.951i)5-s + (0.570 − 0.414i)7-s + (−2.48 − 1.80i)8-s + 1.09·10-s + (−3.31 − 0.189i)11-s + (−1.45 − 4.48i)13-s + (−0.624 − 0.453i)14-s + (−0.543 + 1.67i)16-s + (2.40 − 7.39i)17-s + (0.970 + 0.705i)19-s + (0.247 + 0.761i)20-s + (0.922 + 3.51i)22-s + 6.89·23-s + ⋯
L(s)  = 1  + (−0.239 − 0.736i)2-s + (0.323 − 0.235i)4-s + (−0.138 + 0.425i)5-s + (0.215 − 0.156i)7-s + (−0.877 − 0.637i)8-s + 0.346·10-s + (−0.998 − 0.0572i)11-s + (−0.403 − 1.24i)13-s + (−0.166 − 0.121i)14-s + (−0.135 + 0.418i)16-s + (0.583 − 1.79i)17-s + (0.222 + 0.161i)19-s + (0.0553 + 0.170i)20-s + (0.196 + 0.749i)22-s + 1.43·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.667 + 0.744i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467430 - 1.04750i\)
\(L(\frac12)\) \(\approx\) \(0.467430 - 1.04750i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (3.31 + 0.189i)T \)
good2 \( 1 + (0.338 + 1.04i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-0.570 + 0.414i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.45 + 4.48i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.40 + 7.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.970 - 0.705i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.89T + 23T^{2} \)
29 \( 1 + (-1.07 + 0.780i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.37 + 7.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.82 - 4.95i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.188 + 0.136i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.32T + 43T^{2} \)
47 \( 1 + (-6.73 - 4.89i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.10 + 6.48i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.86 - 2.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.35 - 10.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.04T + 67T^{2} \)
71 \( 1 + (0.207 - 0.637i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.04 + 2.94i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.704 - 2.16i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.652 + 2.00i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.34T + 89T^{2} \)
97 \( 1 + (-1.02 - 3.15i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.65638496081039406979121201271, −9.987243942916100520111688683034, −9.144411635274508660185378619803, −7.71864455871042731669575819309, −7.21605805997038448465279238604, −5.83274507821521217860049929435, −4.96688402297166564010603519964, −3.17191342349397642733055779490, −2.59846562020701620068054668184, −0.72883654270514820066593498704, 1.95890642289436256723684885833, 3.40853993136952424299250723148, 4.86867744821804723154842619794, 5.75862184932043485701800661601, 6.88266268114094675723303699950, 7.59321049045264043980438217203, 8.547661675334773425431480036866, 9.091298508206010887061488102291, 10.46432475239183133250025358829, 11.20240741845388880630808949351

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