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Properties

Label 2-495-11.5-c1-0-3
Degree $2$
Conductor $495$
Sign $-0.988 + 0.154i$
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  + (0.579 + 1.78i)2-s + (−1.22 + 0.893i)4-s + (−0.309 + 0.951i)5-s + (−3.44 + 2.50i)7-s + (0.729 + 0.529i)8-s − 1.87·10-s + (−2.44 − 2.24i)11-s + (0.420 + 1.29i)13-s + (−6.46 − 4.69i)14-s + (−1.46 + 4.49i)16-s + (−0.648 + 1.99i)17-s + (−0.489 − 0.355i)19-s + (−0.469 − 1.44i)20-s + (2.58 − 5.66i)22-s + 4.39·23-s + ⋯
L(s)  = 1  + (0.409 + 1.26i)2-s + (−0.614 + 0.446i)4-s + (−0.138 + 0.425i)5-s + (−1.30 + 0.945i)7-s + (0.257 + 0.187i)8-s − 0.593·10-s + (−0.737 − 0.675i)11-s + (0.116 + 0.358i)13-s + (−1.72 − 1.25i)14-s + (−0.365 + 1.12i)16-s + (−0.157 + 0.484i)17-s + (−0.112 − 0.0815i)19-s + (−0.104 − 0.323i)20-s + (0.550 − 1.20i)22-s + 0.916·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.988 + 0.154i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.988 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0950487 - 1.22605i\)
\(L(\frac12)\) \(\approx\) \(0.0950487 - 1.22605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (2.44 + 2.24i)T \)
good2 \( 1 + (-0.579 - 1.78i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (3.44 - 2.50i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.420 - 1.29i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.648 - 1.99i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.489 + 0.355i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.39T + 23T^{2} \)
29 \( 1 + (5.36 - 3.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.678 + 2.08i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.99 - 3.62i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.99 - 4.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + (2.48 + 1.80i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.05 - 6.32i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.73 + 7.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.75 - 5.41i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + (-1.61 + 4.98i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.584 + 0.424i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.75 - 5.39i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.294 + 0.906i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 1.24T + 89T^{2} \)
97 \( 1 + (-3.56 - 10.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.32180345710292768929331284676, −10.58276716424942175921563179088, −9.343813768666766781781185823492, −8.593275492779396199649893124583, −7.54902032454940150385895674519, −6.65059998653692021884889556064, −5.98488130212836772314090251219, −5.22503411996197236598470357802, −3.74426920170486163352982701414, −2.55480531449590777837926476546, 0.62739843019553103253884322924, 2.38197460642877296967112792569, 3.48923574351933473897084616534, 4.29773418164837940494407702290, 5.45603766387782070791814158676, 6.95837985835780487719608994763, 7.58502714823292303055902064839, 9.144796412978243996491424614049, 9.872109841080833221766513658269, 10.57159074484912742784207387597

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