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Properties

Label 2-495-11.4-c1-0-6
Degree $2$
Conductor $495$
Sign $0.276 + 0.960i$
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  + (−1.12 − 0.817i)2-s + (−0.0207 − 0.0638i)4-s + (−0.809 + 0.587i)5-s + (0.394 + 1.21i)7-s + (−0.888 + 2.73i)8-s + 1.39·10-s + (1.20 − 3.09i)11-s + (1.14 + 0.833i)13-s + (0.548 − 1.68i)14-s + (3.12 − 2.26i)16-s + (4.04 − 2.93i)17-s + (0.0488 − 0.150i)19-s + (0.0543 + 0.0394i)20-s + (−3.87 + 2.49i)22-s + 5.00·23-s + ⋯
L(s)  = 1  + (−0.795 − 0.577i)2-s + (−0.0103 − 0.0319i)4-s + (−0.361 + 0.262i)5-s + (0.149 + 0.459i)7-s + (−0.313 + 0.966i)8-s + 0.439·10-s + (0.362 − 0.931i)11-s + (0.318 + 0.231i)13-s + (0.146 − 0.451i)14-s + (0.780 − 0.567i)16-s + (0.981 − 0.712i)17-s + (0.0112 − 0.0345i)19-s + (0.0121 + 0.00883i)20-s + (−0.827 + 0.531i)22-s + 1.04·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.276 + 0.960i$
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: \((2,\ 495,\ (\ :1/2),\ 0.276 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681803 - 0.513031i\)
\(L(\frac12)\) \(\approx\) \(0.681803 - 0.513031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 3.09i)T \)
good2 \( 1 + (1.12 + 0.817i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.394 - 1.21i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.14 - 0.833i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.04 + 2.93i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0488 + 0.150i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.00T + 23T^{2} \)
29 \( 1 + (1.93 + 5.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.46 + 1.79i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.45 + 4.46i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.34 - 7.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.41T + 43T^{2} \)
47 \( 1 + (-2.54 + 7.82i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.57 - 5.50i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.50 + 7.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.5 + 8.40i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.38T + 67T^{2} \)
71 \( 1 + (5.48 - 3.98i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.67 - 8.23i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.05 - 1.49i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.18 + 5.94i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + (5.18 + 3.76i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.83183835304051969503987746196, −9.834513504495511330192277904190, −9.052486153566917990982865709574, −8.366152420414519850825727296099, −7.35770880013219032158265237588, −6.03650952583121673484834710676, −5.19615366317525750121759089306, −3.62442213242878274541951601250, −2.43168899833391542021053868103, −0.841960646544957807909112908490, 1.20972652051272957310799304131, 3.37800946996720619674200903545, 4.34200598694307553073981287368, 5.67399052477447140842395564184, 7.03700041368460426189867474531, 7.42358693715643817963881896791, 8.467701069834804904118497499116, 9.115079808614634557915554725114, 10.09933859371934479799845742860, 10.86548127360479633140467037647

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