Properties

Label 2-503-1.1-c1-0-14
Degree $2$
Conductor $503$
Sign $1$
Analytic cond. $4.01647$
Root an. cond. $2.00411$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.89·2-s + 1.34·3-s + 1.57·4-s + 4.23·5-s − 2.54·6-s − 3.43·7-s + 0.794·8-s − 1.18·9-s − 8.00·10-s + 5.77·11-s + 2.12·12-s + 1.41·13-s + 6.50·14-s + 5.70·15-s − 4.66·16-s + 5.54·17-s + 2.23·18-s − 2.00·19-s + 6.68·20-s − 4.63·21-s − 10.9·22-s − 3.79·23-s + 1.07·24-s + 12.9·25-s − 2.67·26-s − 5.63·27-s − 5.42·28-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.778·3-s + 0.789·4-s + 1.89·5-s − 1.04·6-s − 1.29·7-s + 0.281·8-s − 0.394·9-s − 2.53·10-s + 1.73·11-s + 0.614·12-s + 0.391·13-s + 1.73·14-s + 1.47·15-s − 1.16·16-s + 1.34·17-s + 0.527·18-s − 0.460·19-s + 1.49·20-s − 1.01·21-s − 2.32·22-s − 0.792·23-s + 0.218·24-s + 2.58·25-s − 0.523·26-s − 1.08·27-s − 1.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(503\)
Sign: $1$
Analytic conductor: \(4.01647\)
Root analytic conductor: \(2.00411\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 503,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.194019940\)
\(L(\frac12)\) \(\approx\) \(1.194019940\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad503 \( 1 - T \)
good2 \( 1 + 1.89T + 2T^{2} \)
3 \( 1 - 1.34T + 3T^{2} \)
5 \( 1 - 4.23T + 5T^{2} \)
7 \( 1 + 3.43T + 7T^{2} \)
11 \( 1 - 5.77T + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
19 \( 1 + 2.00T + 19T^{2} \)
23 \( 1 + 3.79T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 - 4.53T + 31T^{2} \)
37 \( 1 - 4.04T + 37T^{2} \)
41 \( 1 - 6.27T + 41T^{2} \)
43 \( 1 + 0.618T + 43T^{2} \)
47 \( 1 - 4.16T + 47T^{2} \)
53 \( 1 - 6.53T + 53T^{2} \)
59 \( 1 - 6.82T + 59T^{2} \)
61 \( 1 + 5.36T + 61T^{2} \)
67 \( 1 - 0.00814T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 0.481T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 5.05T + 83T^{2} \)
89 \( 1 - 4.64T + 89T^{2} \)
97 \( 1 - 3.07T + 97T^{2} \)
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

−10.25770171365423604602878611724, −9.774838795067279355208670839997, −9.156238838300678167057017005196, −8.810725163370550617562295426821, −7.47292836448779234544990122573, −6.33209981199964438883211585670, −5.85085328247624473701005801188, −3.77404740179310321805743194598, −2.47249056403073583645133424581, −1.33770366547113799831762316796, 1.33770366547113799831762316796, 2.47249056403073583645133424581, 3.77404740179310321805743194598, 5.85085328247624473701005801188, 6.33209981199964438883211585670, 7.47292836448779234544990122573, 8.810725163370550617562295426821, 9.156238838300678167057017005196, 9.774838795067279355208670839997, 10.25770171365423604602878611724

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