Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 503 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2.53·5-s + 3.60·7-s + 9-s + 4.66·11-s − 2.60·13-s − 2.53·15-s + 2.74·17-s + 0.716·19-s − 3.60·21-s + 7.76·23-s + 1.42·25-s − 27-s + 8.07·29-s − 4.58·31-s − 4.66·33-s + 9.13·35-s − 10.4·37-s + 2.60·39-s + 10.0·41-s − 1.83·43-s + 2.53·45-s + 2.79·47-s + 6.00·49-s − 2.74·51-s − 10.2·53-s + 11.8·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·5-s + 1.36·7-s + 0.333·9-s + 1.40·11-s − 0.722·13-s − 0.654·15-s + 0.665·17-s + 0.164·19-s − 0.786·21-s + 1.61·23-s + 0.284·25-s − 0.192·27-s + 1.50·29-s − 0.823·31-s − 0.812·33-s + 1.54·35-s − 1.71·37-s + 0.416·39-s + 1.57·41-s − 0.279·43-s + 0.377·45-s + 0.407·47-s + 0.857·49-s − 0.384·51-s − 1.41·53-s + 1.59·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6036\)    =    \(2^{2} \cdot 3 \cdot 503\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.989472333$
$L(\frac12)$  $\approx$  $2.989472333$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
503 \( 1 + T \)
good5 \( 1 - 2.53T + 5T^{2} \)
7 \( 1 - 3.60T + 7T^{2} \)
11 \( 1 - 4.66T + 11T^{2} \)
13 \( 1 + 2.60T + 13T^{2} \)
17 \( 1 - 2.74T + 17T^{2} \)
19 \( 1 - 0.716T + 19T^{2} \)
23 \( 1 - 7.76T + 23T^{2} \)
29 \( 1 - 8.07T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 1.83T + 43T^{2} \)
47 \( 1 - 2.79T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 5.44T + 59T^{2} \)
61 \( 1 - 8.24T + 61T^{2} \)
67 \( 1 + 1.77T + 67T^{2} \)
71 \( 1 - 0.897T + 71T^{2} \)
73 \( 1 - 0.515T + 73T^{2} \)
79 \( 1 - 6.23T + 79T^{2} \)
83 \( 1 + 7.92T + 83T^{2} \)
89 \( 1 - 3.24T + 89T^{2} \)
97 \( 1 + 0.898T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.069773417046006530412713118723, −7.17506760616929985143666512214, −6.69289412565712386268643432264, −5.83900917917432019861762250169, −5.16833566640890119155995491075, −4.75163049722214564710503475435, −3.75871412777169176573742475446, −2.58912246943978190903541844362, −1.60254348758480473928904293868, −1.07858953317859263842364653778, 1.07858953317859263842364653778, 1.60254348758480473928904293868, 2.58912246943978190903541844362, 3.75871412777169176573742475446, 4.75163049722214564710503475435, 5.16833566640890119155995491075, 5.83900917917432019861762250169, 6.69289412565712386268643432264, 7.17506760616929985143666512214, 8.069773417046006530412713118723

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