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Properties

Label 2-495-15.14-c2-0-21
Degree $2$
Conductor $495$
Sign $-0.499 + 0.866i$
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  − 3.50·2-s + 8.27·4-s + (−4.54 + 2.09i)5-s + 0.502i·7-s − 14.9·8-s + (15.9 − 7.33i)10-s + 3.31i·11-s + 8.22i·13-s − 1.75i·14-s + 19.3·16-s − 10.4·17-s − 3.26·19-s + (−37.5 + 17.3i)20-s − 11.6i·22-s + 14.7·23-s + ⋯
L(s)  = 1  − 1.75·2-s + 2.06·4-s + (−0.908 + 0.418i)5-s + 0.0717i·7-s − 1.87·8-s + (1.59 − 0.733i)10-s + 0.301i·11-s + 0.632i·13-s − 0.125i·14-s + 1.20·16-s − 0.612·17-s − 0.171·19-s + (−1.87 + 0.866i)20-s − 0.528i·22-s + 0.641·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.499 + 0.866i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ -0.499 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1168096148\)
\(L(\frac12)\) \(\approx\) \(0.1168096148\)
\(L(2)\) 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 \( 1 + (4.54 - 2.09i)T \)
11 \( 1 - 3.31iT \)
good2 \( 1 + 3.50T + 4T^{2} \)
7 \( 1 - 0.502iT - 49T^{2} \)
13 \( 1 - 8.22iT - 169T^{2} \)
17 \( 1 + 10.4T + 289T^{2} \)
19 \( 1 + 3.26T + 361T^{2} \)
23 \( 1 - 14.7T + 529T^{2} \)
29 \( 1 - 24.2iT - 841T^{2} \)
31 \( 1 + 26.7T + 961T^{2} \)
37 \( 1 - 29.1iT - 1.36e3T^{2} \)
41 \( 1 + 27.6iT - 1.68e3T^{2} \)
43 \( 1 - 27.6iT - 1.84e3T^{2} \)
47 \( 1 - 21.3T + 2.20e3T^{2} \)
53 \( 1 + 21.4T + 2.80e3T^{2} \)
59 \( 1 + 53.5iT - 3.48e3T^{2} \)
61 \( 1 + 120.T + 3.72e3T^{2} \)
67 \( 1 + 108. iT - 4.48e3T^{2} \)
71 \( 1 + 115. iT - 5.04e3T^{2} \)
73 \( 1 + 129. iT - 5.32e3T^{2} \)
79 \( 1 + 80.7T + 6.24e3T^{2} \)
83 \( 1 + 112.T + 6.88e3T^{2} \)
89 \( 1 + 67.6iT - 7.92e3T^{2} \)
97 \( 1 - 78.3iT - 9.40e3T^{2} \)
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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

−10.51054969265130596423301695501, −9.308871217671721639225905761217, −8.800332164862020056648604850678, −7.78940932236709777990261308059, −7.13648311198734294733109231704, −6.37011213752929271992336807345, −4.58838990399483765918678485225, −3.11036323152667665862638314470, −1.75270587991652996529823966532, −0.099846054267205017184010987737, 1.06879929564586935736882323388, 2.67542358449755228735990926823, 4.14556344163508790208609100929, 5.67881092924903262941863116422, 7.00330586035571132502204592111, 7.62150437433646664480349256306, 8.531576720654135510311637481991, 9.003658975047275960057108564280, 10.05925507840384062043324387519, 10.93337014910446616975809566062

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