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Properties

Label 2-495-11.4-c1-0-19
Degree $2$
Conductor $495$
Sign $-0.884 + 0.467i$
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  + (−0.212 − 0.154i)2-s + (−0.596 − 1.83i)4-s + (0.809 − 0.587i)5-s + (−0.986 − 3.03i)7-s + (−0.318 + 0.980i)8-s − 0.262·10-s + (−3.27 − 0.547i)11-s + (0.905 + 0.658i)13-s + (−0.258 + 0.796i)14-s + (−2.90 + 2.11i)16-s + (−0.0713 + 0.0518i)17-s + (−0.0212 + 0.0654i)19-s + (−1.56 − 1.13i)20-s + (0.609 + 0.620i)22-s − 6.65·23-s + ⋯
L(s)  = 1  + (−0.150 − 0.109i)2-s + (−0.298 − 0.918i)4-s + (0.361 − 0.262i)5-s + (−0.372 − 1.14i)7-s + (−0.112 + 0.346i)8-s − 0.0829·10-s + (−0.986 − 0.165i)11-s + (0.251 + 0.182i)13-s + (−0.0691 + 0.212i)14-s + (−0.726 + 0.527i)16-s + (−0.0173 + 0.0125i)17-s + (−0.00487 + 0.0150i)19-s + (−0.349 − 0.253i)20-s + (0.130 + 0.132i)22-s − 1.38·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.884 + 0.467i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199409 - 0.803866i\)
\(L(\frac12)\) \(\approx\) \(0.199409 - 0.803866i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (3.27 + 0.547i)T \)
good2 \( 1 + (0.212 + 0.154i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.986 + 3.03i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.905 - 0.658i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0713 - 0.0518i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0212 - 0.0654i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.65T + 23T^{2} \)
29 \( 1 + (1.15 + 3.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.75 - 5.63i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.57 + 7.92i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.60 + 11.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + (0.280 - 0.863i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.705 - 0.512i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.567 + 1.74i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.13 + 5.91i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.53T + 67T^{2} \)
71 \( 1 + (3.77 - 2.74i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.21 + 6.80i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.640 + 0.465i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.200 + 0.145i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + (-3.31 - 2.40i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.23227820874030582939625570025, −10.07219005365280803312266721151, −8.895359134160342540650676377470, −7.955824856844631832503432613821, −6.78500778785077872037979844693, −5.85221819089097321745086929482, −4.90304786865758778521973105232, −3.79226459359172011655807744786, −2.06462993139650954837751504465, −0.50394352307388378666981602917, 2.38518640892203230353886079196, 3.25781397585747330690609083637, 4.69105592222094969319408571844, 5.81531363194089514054447035679, 6.72693575657578000948686893251, 8.003683107281503484129246633823, 8.438902487548763121777086938612, 9.581318182426259192483357475953, 10.17131297896532015294759711555, 11.55306126810301534388957622655

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