Properties

Label 2-495-11.5-c1-0-2
Degree $2$
Conductor $495$
Sign $-0.246 + 0.969i$
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  + (0.850 + 2.61i)2-s + (−4.51 + 3.27i)4-s + (0.309 − 0.951i)5-s + (−2.21 + 1.60i)7-s + (−7.96 − 5.78i)8-s + 2.75·10-s + (−3.02 + 1.36i)11-s + (−0.857 − 2.63i)13-s + (−6.08 − 4.41i)14-s + (4.92 − 15.1i)16-s + (−1.16 + 3.59i)17-s + (3.43 + 2.49i)19-s + (1.72 + 5.30i)20-s + (−6.14 − 6.74i)22-s − 1.37·23-s + ⋯
L(s)  = 1  + (0.601 + 1.85i)2-s + (−2.25 + 1.63i)4-s + (0.138 − 0.425i)5-s + (−0.835 + 0.606i)7-s + (−2.81 − 2.04i)8-s + 0.870·10-s + (−0.911 + 0.412i)11-s + (−0.237 − 0.731i)13-s + (−1.62 − 1.18i)14-s + (1.23 − 3.78i)16-s + (−0.283 + 0.872i)17-s + (0.789 + 0.573i)19-s + (0.385 + 1.18i)20-s + (−1.31 − 1.43i)22-s − 0.287·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.246 + 0.969i$
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: \((2,\ 495,\ (\ :1/2),\ -0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503334 - 0.647573i\)
\(L(\frac12)\) \(\approx\) \(0.503334 - 0.647573i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.02 - 1.36i)T \)
good2 \( 1 + (-0.850 - 2.61i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (2.21 - 1.60i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.857 + 2.63i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.16 - 3.59i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.43 - 2.49i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.37T + 23T^{2} \)
29 \( 1 + (7.17 - 5.21i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.68 - 5.19i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.86 + 1.35i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.97 + 5.06i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + (-4.52 - 3.28i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.168 + 0.517i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.314 - 0.228i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.43 - 13.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.65T + 67T^{2} \)
71 \( 1 + (2.83 - 8.71i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.60 - 4.79i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.06 + 3.27i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.43 - 4.42i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.62T + 89T^{2} \)
97 \( 1 + (5.12 + 15.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.25457064714463254264436103089, −10.36104966497727835756751663821, −9.377772322893272237245819538531, −8.642720237824115795363047337518, −7.74583430077525950830454051810, −7.00919515774844568671082092764, −5.67552762711046873061922153529, −5.58281332305783941142825255633, −4.25350468738234309837349362985, −3.06514970826954468580048257686, 0.39256072666732757350151535913, 2.26797967690494632961977346026, 3.13848006221248496232511058642, 4.12748826099058286289285233755, 5.19018148770830367738276269737, 6.27328887929246042754871291421, 7.66013779460898755247426866541, 9.245925324045435684520871159713, 9.660747787994206893103959864043, 10.50047370321484208082449541392

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