Properties

Label 2-495-1.1-c1-0-11
Degree $2$
Conductor $495$
Sign $-1$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s + 3.82·4-s + 5-s − 2·7-s − 4.41·8-s − 2.41·10-s − 11-s − 1.17·13-s + 4.82·14-s + 2.99·16-s − 6.82·17-s + 3.82·20-s + 2.41·22-s + 2.82·23-s + 25-s + 2.82·26-s − 7.65·28-s + 3.65·29-s + 1.58·32-s + 16.4·34-s − 2·35-s − 7.65·37-s − 4.41·40-s − 6·41-s − 6·43-s − 3.82·44-s − 6.82·46-s + ⋯
L(s)  = 1  − 1.70·2-s + 1.91·4-s + 0.447·5-s − 0.755·7-s − 1.56·8-s − 0.763·10-s − 0.301·11-s − 0.324·13-s + 1.29·14-s + 0.749·16-s − 1.65·17-s + 0.856·20-s + 0.514·22-s + 0.589·23-s + 0.200·25-s + 0.554·26-s − 1.44·28-s + 0.679·29-s + 0.280·32-s + 2.82·34-s − 0.338·35-s − 1.25·37-s − 0.697·40-s − 0.937·41-s − 0.914·43-s − 0.577·44-s − 1.00·46-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
good2 \( 1 + 2.41T + 2T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−10.32470334397441228459577658957, −9.504751231990373787742277824046, −8.899182679847254668894049577711, −8.042118646857261565031079837643, −6.86104477989725159190689622839, −6.45030383076292275925719248690, −4.88419799925923978199403676724, −3.01478731580852465028396046576, −1.83340295695726343730774282839, 0, 1.83340295695726343730774282839, 3.01478731580852465028396046576, 4.88419799925923978199403676724, 6.45030383076292275925719248690, 6.86104477989725159190689622839, 8.042118646857261565031079837643, 8.899182679847254668894049577711, 9.504751231990373787742277824046, 10.32470334397441228459577658957

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