Properties

Label 2-495-1.1-c1-0-2
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 1.82·4-s + 5-s − 4.82·7-s + 1.58·8-s − 0.414·10-s + 11-s + 5.65·13-s + 1.99·14-s + 3·16-s + 6.82·17-s − 1.17·19-s − 1.82·20-s − 0.414·22-s + 4·23-s + 25-s − 2.34·26-s + 8.82·28-s − 0.828·29-s − 4.41·32-s − 2.82·34-s − 4.82·35-s + 0.343·37-s + 0.485·38-s + 1.58·40-s + 0.828·41-s − 3.17·43-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s + 0.447·5-s − 1.82·7-s + 0.560·8-s − 0.130·10-s + 0.301·11-s + 1.56·13-s + 0.534·14-s + 0.750·16-s + 1.65·17-s − 0.268·19-s − 0.408·20-s − 0.0883·22-s + 0.834·23-s + 0.200·25-s − 0.459·26-s + 1.66·28-s − 0.153·29-s − 0.780·32-s − 0.485·34-s − 0.816·35-s + 0.0564·37-s + 0.0787·38-s + 0.250·40-s + 0.129·41-s − 0.483·43-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

\(L(1)\) \(\approx\) \(0.9657007276\)
\(L(\frac12)\) \(\approx\) \(0.9657007276\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 + 4.82T + 7T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.343T + 37T^{2} \)
41 \( 1 - 0.828T + 41T^{2} \)
43 \( 1 + 3.17T + 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 0.343T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51843968414512898332519903296, −10.00169364499106846710884884196, −9.169724094385808836303108789430, −8.602383164944062514213902740200, −7.29379222370783295255333182248, −6.20289516130001992071945017691, −5.52114127546218421109580391860, −3.92490376183820598138637798674, −3.18629491234324629698620261642, −0.992081852202220186506946909778, 0.992081852202220186506946909778, 3.18629491234324629698620261642, 3.92490376183820598138637798674, 5.52114127546218421109580391860, 6.20289516130001992071945017691, 7.29379222370783295255333182248, 8.602383164944062514213902740200, 9.169724094385808836303108789430, 10.00169364499106846710884884196, 10.51843968414512898332519903296

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