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Properties

Label 2-495-5.3-c2-0-7
Degree $2$
Conductor $495$
Sign $-0.852 - 0.523i$
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  + (−0.364 + 0.364i)2-s + 3.73i·4-s + (−0.0133 − 4.99i)5-s + (−2.00 + 2.00i)7-s + (−2.82 − 2.82i)8-s + (1.82 + 1.81i)10-s + 3.31·11-s + (3.51 + 3.51i)13-s − 1.45i·14-s − 12.8·16-s + (−8.30 + 8.30i)17-s + 19.7i·19-s + (18.6 − 0.0498i)20-s + (−1.20 + 1.20i)22-s + (2.95 + 2.95i)23-s + ⋯
L(s)  = 1  + (−0.182 + 0.182i)2-s + 0.933i·4-s + (−0.00266 − 0.999i)5-s + (−0.285 + 0.285i)7-s + (−0.352 − 0.352i)8-s + (0.182 + 0.181i)10-s + 0.301·11-s + (0.270 + 0.270i)13-s − 0.104i·14-s − 0.804·16-s + (−0.488 + 0.488i)17-s + 1.04i·19-s + (0.933 − 0.00249i)20-s + (−0.0549 + 0.0549i)22-s + (0.128 + 0.128i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.852 - 0.523i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ -0.852 - 0.523i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7503859961\)
\(L(\frac12)\) \(\approx\) \(0.7503859961\)
\(L(2)\) 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 \( 1 + (0.0133 + 4.99i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 + (0.364 - 0.364i)T - 4iT^{2} \)
7 \( 1 + (2.00 - 2.00i)T - 49iT^{2} \)
13 \( 1 + (-3.51 - 3.51i)T + 169iT^{2} \)
17 \( 1 + (8.30 - 8.30i)T - 289iT^{2} \)
19 \( 1 - 19.7iT - 361T^{2} \)
23 \( 1 + (-2.95 - 2.95i)T + 529iT^{2} \)
29 \( 1 - 33.7iT - 841T^{2} \)
31 \( 1 + 35.5T + 961T^{2} \)
37 \( 1 + (20.7 - 20.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 10.5T + 1.68e3T^{2} \)
43 \( 1 + (25.7 + 25.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-8.30 + 8.30i)T - 2.20e3iT^{2} \)
53 \( 1 + (29.2 + 29.2i)T + 2.80e3iT^{2} \)
59 \( 1 - 17.0iT - 3.48e3T^{2} \)
61 \( 1 - 2.32T + 3.72e3T^{2} \)
67 \( 1 + (55.1 - 55.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 82.4T + 5.04e3T^{2} \)
73 \( 1 + (-57.3 - 57.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 87.8iT - 6.24e3T^{2} \)
83 \( 1 + (109. + 109. i)T + 6.88e3iT^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + (-49.8 + 49.8i)T - 9.40e3iT^{2} \)
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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

−11.28445999576800955220395366914, −10.02870958447395466598086827290, −8.935380623170534083267797845050, −8.644325901921206904010557315956, −7.60927842430370769829205444397, −6.59993088804632363400904530645, −5.50601894436173548953341030664, −4.26917284338830834404628904574, −3.37800309698389135583629240752, −1.69581321254829412225300453512, 0.30941862008468422564632026330, 2.04048135987202070510893779391, 3.23753336389687487911550991314, 4.61353246343015185926827103174, 5.85326140462776334165215057507, 6.63983695919333062750232833268, 7.43615385237699065723322638856, 8.846511723288488069985786670313, 9.623321621250246415078016405159, 10.43211675309938307871088088329

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