Properties

Label 2-507-1.1-c3-0-13
Degree $2$
Conductor $507$
Sign $1$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.36·2-s − 3·3-s − 2.42·4-s + 6.42·5-s − 7.08·6-s − 29.4·7-s − 24.6·8-s + 9·9-s + 15.1·10-s − 0.624·11-s + 7.26·12-s − 69.6·14-s − 19.2·15-s − 38.7·16-s + 87.7·17-s + 21.2·18-s + 82.8·19-s − 15.5·20-s + 88.4·21-s − 1.47·22-s − 74.7·23-s + 73.8·24-s − 83.7·25-s − 27·27-s + 71.4·28-s + 226.·29-s − 45.5·30-s + ⋯
L(s)  = 1  + 0.835·2-s − 0.577·3-s − 0.302·4-s + 0.574·5-s − 0.482·6-s − 1.59·7-s − 1.08·8-s + 0.333·9-s + 0.479·10-s − 0.0171·11-s + 0.174·12-s − 1.32·14-s − 0.331·15-s − 0.605·16-s + 1.25·17-s + 0.278·18-s + 0.999·19-s − 0.173·20-s + 0.919·21-s − 0.0142·22-s − 0.678·23-s + 0.628·24-s − 0.670·25-s − 0.192·27-s + 0.482·28-s + 1.44·29-s − 0.276·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.685016938\)
\(L(\frac12)\) \(\approx\) \(1.685016938\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 - 2.36T + 8T^{2} \)
5 \( 1 - 6.42T + 125T^{2} \)
7 \( 1 + 29.4T + 343T^{2} \)
11 \( 1 + 0.624T + 1.33e3T^{2} \)
17 \( 1 - 87.7T + 4.91e3T^{2} \)
19 \( 1 - 82.8T + 6.85e3T^{2} \)
23 \( 1 + 74.7T + 1.21e4T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 - 173.T + 2.97e4T^{2} \)
37 \( 1 - 112.T + 5.06e4T^{2} \)
41 \( 1 + 267.T + 6.89e4T^{2} \)
43 \( 1 - 383.T + 7.95e4T^{2} \)
47 \( 1 - 337.T + 1.03e5T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 + 529.T + 2.05e5T^{2} \)
61 \( 1 - 203.T + 2.26e5T^{2} \)
67 \( 1 - 121.T + 3.00e5T^{2} \)
71 \( 1 + 661.T + 3.57e5T^{2} \)
73 \( 1 - 167.T + 3.89e5T^{2} \)
79 \( 1 + 101.T + 4.93e5T^{2} \)
83 \( 1 + 506.T + 5.71e5T^{2} \)
89 \( 1 - 1.40e3T + 7.04e5T^{2} \)
97 \( 1 - 1.90e3T + 9.12e5T^{2} \)
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

−10.14995921365836295370013161200, −9.902550313303509560237305585432, −8.977523093989725465056687339119, −7.57098520382177320938106994553, −6.22728171331706402681004583156, −5.97755474265149850402812630461, −4.90561678015233118871595437802, −3.70287919248222052545906968165, −2.82373323396071499020193185827, −0.72386101712323911053632574865, 0.72386101712323911053632574865, 2.82373323396071499020193185827, 3.70287919248222052545906968165, 4.90561678015233118871595437802, 5.97755474265149850402812630461, 6.22728171331706402681004583156, 7.57098520382177320938106994553, 8.977523093989725465056687339119, 9.902550313303509560237305585432, 10.14995921365836295370013161200

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