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 − 2.38·5-s + 0.588·7-s + 9-s − 6.40·11-s − 3.59·13-s + 2.38·15-s − 3.38·17-s + 7.29·19-s − 0.588·21-s + 0.312·23-s + 0.711·25-s − 27-s − 5.17·29-s − 0.272·31-s + 6.40·33-s − 1.40·35-s − 2.33·37-s + 3.59·39-s − 1.99·41-s − 4.73·43-s − 2.38·45-s − 9.43·47-s − 6.65·49-s + 3.38·51-s − 12.3·53-s + 15.2·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.06·5-s + 0.222·7-s + 0.333·9-s − 1.92·11-s − 0.996·13-s + 0.617·15-s − 0.820·17-s + 1.67·19-s − 0.128·21-s + 0.0650·23-s + 0.142·25-s − 0.192·27-s − 0.960·29-s − 0.0488·31-s + 1.11·33-s − 0.237·35-s − 0.384·37-s + 0.575·39-s − 0.312·41-s − 0.721·43-s − 0.356·45-s − 1.37·47-s − 0.950·49-s + 0.473·51-s − 1.69·53-s + 2.06·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3318439720$
$L(\frac12)$  $\approx$  $0.3318439720$
$L(\frac{3}{2})$   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 \( 1 + 2.38T + 5T^{2} \)
7 \( 1 - 0.588T + 7T^{2} \)
11 \( 1 + 6.40T + 11T^{2} \)
13 \( 1 + 3.59T + 13T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 - 7.29T + 19T^{2} \)
23 \( 1 - 0.312T + 23T^{2} \)
29 \( 1 + 5.17T + 29T^{2} \)
31 \( 1 + 0.272T + 31T^{2} \)
37 \( 1 + 2.33T + 37T^{2} \)
41 \( 1 + 1.99T + 41T^{2} \)
43 \( 1 + 4.73T + 43T^{2} \)
47 \( 1 + 9.43T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 6.36T + 59T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 + 9.64T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 - 2.81T + 73T^{2} \)
79 \( 1 - 2.41T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 9.03T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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

−7.87459852141497428914483838341, −7.47642751890250217156077366029, −6.87521902238424694382591235589, −5.77430357098876992515046763057, −4.92516235953182730993321508192, −4.85660462171357136738168228997, −3.59903592529652690982500175018, −2.89069768220925113812069303943, −1.83467024988926285653657653341, −0.29657642886246120847937450251, 0.29657642886246120847937450251, 1.83467024988926285653657653341, 2.89069768220925113812069303943, 3.59903592529652690982500175018, 4.85660462171357136738168228997, 4.92516235953182730993321508192, 5.77430357098876992515046763057, 6.87521902238424694382591235589, 7.47642751890250217156077366029, 7.87459852141497428914483838341

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