Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.78·5-s − 2.07·7-s + 9-s + 2.20·11-s − 6.61·13-s + 1.78·15-s − 6.93·17-s − 3.40·19-s + 2.07·21-s + 2.31·23-s − 1.81·25-s − 27-s − 1.77·29-s + 5.22·31-s − 2.20·33-s + 3.71·35-s − 10.8·37-s + 6.61·39-s + 3.10·41-s − 3.95·43-s − 1.78·45-s + 6.71·47-s − 2.67·49-s + 6.93·51-s − 9.33·53-s − 3.93·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.798·5-s − 0.785·7-s + 0.333·9-s + 0.664·11-s − 1.83·13-s + 0.461·15-s − 1.68·17-s − 0.781·19-s + 0.453·21-s + 0.483·23-s − 0.362·25-s − 0.192·27-s − 0.329·29-s + 0.938·31-s − 0.383·33-s + 0.627·35-s − 1.78·37-s + 1.05·39-s + 0.485·41-s − 0.603·43-s − 0.266·45-s + 0.978·47-s − 0.382·49-s + 0.971·51-s − 1.28·53-s − 0.530·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2557355789$
$L(\frac12)$  $\approx$  $0.2557355789$
$L(\frac{3}{2})$   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 \( 1 + 1.78T + 5T^{2} \)
7 \( 1 + 2.07T + 7T^{2} \)
11 \( 1 - 2.20T + 11T^{2} \)
13 \( 1 + 6.61T + 13T^{2} \)
17 \( 1 + 6.93T + 17T^{2} \)
19 \( 1 + 3.40T + 19T^{2} \)
23 \( 1 - 2.31T + 23T^{2} \)
29 \( 1 + 1.77T + 29T^{2} \)
31 \( 1 - 5.22T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 - 3.10T + 41T^{2} \)
43 \( 1 + 3.95T + 43T^{2} \)
47 \( 1 - 6.71T + 47T^{2} \)
53 \( 1 + 9.33T + 53T^{2} \)
59 \( 1 + 8.55T + 59T^{2} \)
61 \( 1 - 8.78T + 61T^{2} \)
67 \( 1 + 8.71T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 9.60T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 3.56T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.924602644873091374368669120691, −7.22397036082009629567655370132, −6.67949149978608698410697313485, −6.16431771472366048600713327562, −4.99602051170556647505776540639, −4.52127099770632993391397221519, −3.78279911659493265783682977152, −2.80017058462752166558248816448, −1.87318499919081933578392483283, −0.25799222473851706727821619090, 0.25799222473851706727821619090, 1.87318499919081933578392483283, 2.80017058462752166558248816448, 3.78279911659493265783682977152, 4.52127099770632993391397221519, 4.99602051170556647505776540639, 6.16431771472366048600713327562, 6.67949149978608698410697313485, 7.22397036082009629567655370132, 7.924602644873091374368669120691

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