Properties

Label 2-495-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 0.999·4-s + 5-s + 2·7-s − 1.73·8-s + 1.73·10-s + 11-s + 5.46·13-s + 3.46·14-s − 5·16-s + 5.46·19-s + 0.999·20-s + 1.73·22-s − 6.92·23-s + 25-s + 9.46·26-s + 1.99·28-s + 3.46·29-s − 10.9·31-s − 5.19·32-s + 2·35-s − 4.92·37-s + 9.46·38-s − 1.73·40-s − 3.46·41-s − 4.92·43-s + 0.999·44-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.499·4-s + 0.447·5-s + 0.755·7-s − 0.612·8-s + 0.547·10-s + 0.301·11-s + 1.51·13-s + 0.925·14-s − 1.25·16-s + 1.25·19-s + 0.223·20-s + 0.369·22-s − 1.44·23-s + 0.200·25-s + 1.85·26-s + 0.377·28-s + 0.643·29-s − 1.96·31-s − 0.918·32-s + 0.338·35-s − 0.810·37-s + 1.53·38-s − 0.273·40-s − 0.541·41-s − 0.751·43-s + 0.150·44-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: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.841229699\)
\(L(\frac12)\) \(\approx\) \(2.841229699\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + 4.92T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 4.92T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 0.928T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 8.39T + 73T^{2} \)
79 \( 1 + 6.53T + 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−11.30304771462910117059273356339, −10.20272727305579513766541437459, −9.084569257831171498526330273772, −8.312340729611254082072872823662, −7.00778467808788486620120121749, −5.90897232901399477075700043561, −5.35119451808805783258496348533, −4.17158148826676126371529980173, −3.31455491747336655126805878644, −1.69968450584551669242086668275, 1.69968450584551669242086668275, 3.31455491747336655126805878644, 4.17158148826676126371529980173, 5.35119451808805783258496348533, 5.90897232901399477075700043561, 7.00778467808788486620120121749, 8.312340729611254082072872823662, 9.084569257831171498526330273772, 10.20272727305579513766541437459, 11.30304771462910117059273356339

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