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Properties

Label 2-495-55.43-c1-0-26
Degree $2$
Conductor $495$
Sign $-0.960 + 0.278i$
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  − 2i·4-s + (−1.65 − 1.5i)5-s − 3.31·11-s − 4·16-s + (−3 + 3.31i)20-s + (−2.84 − 2.84i)23-s + (0.5 + 4.97i)25-s − 9.94·31-s + (1.47 − 1.47i)37-s + 6.63i·44-s + (9.31 − 9.31i)47-s − 7i·49-s + (−3.63 − 3.63i)53-s + (5.5 + 4.97i)55-s − 3.31i·59-s + ⋯
L(s)  = 1  − i·4-s + (−0.741 − 0.670i)5-s − 1.00·11-s − 16-s + (−0.670 + 0.741i)20-s + (−0.592 − 0.592i)23-s + (0.100 + 0.994i)25-s − 1.78·31-s + (0.242 − 0.242i)37-s + 1.00i·44-s + (1.35 − 1.35i)47-s − i·49-s + (−0.499 − 0.499i)53-s + (0.741 + 0.670i)55-s − 0.431i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.960 + 0.278i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.960 + 0.278i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0910265 - 0.641507i\)
\(L(\frac12)\) \(\approx\) \(0.0910265 - 0.641507i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 1.5i)T \)
11 \( 1 + 3.31T \)
good2 \( 1 + 2iT^{2} \)
7 \( 1 + 7iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (2.84 + 2.84i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 9.94T + 31T^{2} \)
37 \( 1 + (-1.47 + 1.47i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-9.31 + 9.31i)T - 47iT^{2} \)
53 \( 1 + (3.63 + 3.63i)T + 53iT^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-11.4 + 11.4i)T - 67iT^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 + (3.52 - 3.52i)T - 97iT^{2} \)
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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

−10.61487623734222731368939366905, −9.679892057795314060469202134481, −8.780980871958665361482168744964, −7.889812421802828281950861730757, −6.88760561470218143496508899027, −5.61177083109721757128607903966, −4.96966564952909587647106592625, −3.80130001666780003834059531876, −2.07804869394233231739161269503, −0.37048664922876438918029661543, 2.49581096514771664488724975540, 3.47405031030584550676182254106, 4.42667115387506240720446252366, 5.81890170407889759323750397280, 7.15272354519761523863634944298, 7.64815083755940574382124367817, 8.437261427138818851199041521544, 9.533139802330753181566372660677, 10.73574870516769500610637609096, 11.27421629407851016249155192694

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