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Properties

Label 2-495-5.2-c2-0-39
Degree $2$
Conductor $495$
Sign $0.506 + 0.862i$
Analytic cond. $13.4877$
Root an. cond. $3.67257$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 1.40i)2-s − 0.0494i·4-s + (−4.99 − 0.112i)5-s + (1.33 + 1.33i)7-s + (5.69 − 5.69i)8-s + (−6.86 − 7.18i)10-s − 3.31·11-s + (2.52 − 2.52i)13-s + 3.74i·14-s + 15.7·16-s + (−19.6 − 19.6i)17-s − 24.1i·19-s + (−0.00554 + 0.247i)20-s + (−4.66 − 4.66i)22-s + (−1.52 + 1.52i)23-s + ⋯
L(s)  = 1  + (0.702 + 0.702i)2-s − 0.0123i·4-s + (−0.999 − 0.0224i)5-s + (0.190 + 0.190i)7-s + (0.711 − 0.711i)8-s + (−0.686 − 0.718i)10-s − 0.301·11-s + (0.194 − 0.194i)13-s + 0.267i·14-s + 0.987·16-s + (−1.15 − 1.15i)17-s − 1.26i·19-s + (−0.000277 + 0.0123i)20-s + (−0.211 − 0.211i)22-s + (−0.0662 + 0.0662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ 0.506 + 0.862i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.677165996\)
\(L(\frac12)\) \(\approx\) \(1.677165996\)
\(L(2)\) 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 \( 1 + (4.99 + 0.112i)T \)
11 \( 1 + 3.31T \)
good2 \( 1 + (-1.40 - 1.40i)T + 4iT^{2} \)
7 \( 1 + (-1.33 - 1.33i)T + 49iT^{2} \)
13 \( 1 + (-2.52 + 2.52i)T - 169iT^{2} \)
17 \( 1 + (19.6 + 19.6i)T + 289iT^{2} \)
19 \( 1 + 24.1iT - 361T^{2} \)
23 \( 1 + (1.52 - 1.52i)T - 529iT^{2} \)
29 \( 1 + 52.1iT - 841T^{2} \)
31 \( 1 + 5.66T + 961T^{2} \)
37 \( 1 + (-26.1 - 26.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 0.868T + 1.68e3T^{2} \)
43 \( 1 + (-55.2 + 55.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-7.21 - 7.21i)T + 2.20e3iT^{2} \)
53 \( 1 + (34.7 - 34.7i)T - 2.80e3iT^{2} \)
59 \( 1 - 78.6iT - 3.48e3T^{2} \)
61 \( 1 + 71.8T + 3.72e3T^{2} \)
67 \( 1 + (7.85 + 7.85i)T + 4.48e3iT^{2} \)
71 \( 1 + 75.2T + 5.04e3T^{2} \)
73 \( 1 + (-11.5 + 11.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 132. iT - 6.24e3T^{2} \)
83 \( 1 + (37.5 - 37.5i)T - 6.88e3iT^{2} \)
89 \( 1 - 97.5iT - 7.92e3T^{2} \)
97 \( 1 + (58.4 + 58.4i)T + 9.40e3iT^{2} \)
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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

−10.84344757138322883246772717047, −9.621099988315895047548644023795, −8.636131807441902242894185184046, −7.55612572042915060003775381827, −6.95094134608315819359009192351, −5.87022199752686274511483821700, −4.77439438360729986605050996841, −4.19450240499650472260730810115, −2.66952789542247841364002289408, −0.54181459686011291665947039003, 1.69303840279758658952891134367, 3.14712890491844116902799990531, 4.03448637644994328038041581170, 4.73456525489251704069294898352, 6.11859612610520067013988821330, 7.43720565976253642098100719975, 8.099076131758883793416573166685, 8.978658985227595510718835410337, 10.59069996566599835668655109461, 10.92883622176517081421151947552

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