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Properties

Label 2-495-15.14-c2-0-16
Degree $2$
Conductor $495$
Sign $0.361 - 0.932i$
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  + 2.18·2-s + 0.770·4-s + (4.16 − 2.76i)5-s + 12.4i·7-s − 7.05·8-s + (9.10 − 6.03i)10-s + 3.31i·11-s + 24.4i·13-s + 27.1i·14-s − 18.4·16-s − 16.2·17-s + 27.6·19-s + (3.21 − 2.12i)20-s + 7.24i·22-s + 30.9·23-s + ⋯
L(s)  = 1  + 1.09·2-s + 0.192·4-s + (0.833 − 0.552i)5-s + 1.77i·7-s − 0.881·8-s + (0.910 − 0.603i)10-s + 0.301i·11-s + 1.88i·13-s + 1.93i·14-s − 1.15·16-s − 0.953·17-s + 1.45·19-s + (0.160 − 0.106i)20-s + 0.329i·22-s + 1.34·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.361 - 0.932i$
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: \((2,\ 495,\ (\ :1),\ 0.361 - 0.932i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.888651593\)
\(L(\frac12)\) \(\approx\) \(2.888651593\)
\(L(2)\) 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 \( 1 + (-4.16 + 2.76i)T \)
11 \( 1 - 3.31iT \)
good2 \( 1 - 2.18T + 4T^{2} \)
7 \( 1 - 12.4iT - 49T^{2} \)
13 \( 1 - 24.4iT - 169T^{2} \)
17 \( 1 + 16.2T + 289T^{2} \)
19 \( 1 - 27.6T + 361T^{2} \)
23 \( 1 - 30.9T + 529T^{2} \)
29 \( 1 + 27.8iT - 841T^{2} \)
31 \( 1 - 1.23T + 961T^{2} \)
37 \( 1 - 26.8iT - 1.36e3T^{2} \)
41 \( 1 - 1.58iT - 1.68e3T^{2} \)
43 \( 1 - 26.8iT - 1.84e3T^{2} \)
47 \( 1 + 47.4T + 2.20e3T^{2} \)
53 \( 1 - 61.3T + 2.80e3T^{2} \)
59 \( 1 + 11.6iT - 3.48e3T^{2} \)
61 \( 1 + 7.85T + 3.72e3T^{2} \)
67 \( 1 + 53.1iT - 4.48e3T^{2} \)
71 \( 1 + 23.2iT - 5.04e3T^{2} \)
73 \( 1 - 17.2iT - 5.32e3T^{2} \)
79 \( 1 + 118.T + 6.24e3T^{2} \)
83 \( 1 - 89.5T + 6.88e3T^{2} \)
89 \( 1 + 18.1iT - 7.92e3T^{2} \)
97 \( 1 + 135. iT - 9.40e3T^{2} \)
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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

−11.54180964053495138883514000209, −9.597835080482192774350817174493, −9.223588889311965499667971573037, −8.571531475874972785297137118062, −6.75825591155335827936595190960, −6.01321583330629742172495806909, −5.11851169708004056742277940409, −4.52151888870076508812942895903, −2.90095093021279951887731893297, −1.91388205501383038080354139279, 0.838526901659192990011431927117, 2.95055783989065308983069385552, 3.60583521357670902845945685989, 4.93123413297810250397626993854, 5.63153119763776216080079279123, 6.79403209325631111786870313990, 7.47105554227494801401132957965, 8.861336993832233511096554694256, 9.955684645369271812289747038941, 10.64221737619585066595219654830

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