Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5·7-s − 2·9-s − 5·11-s + 13-s − 4·17-s − 4·19-s + 5·21-s − 9·23-s − 5·25-s + 5·27-s + 2·29-s + 2·31-s + 5·33-s − 6·37-s − 39-s + 6·41-s − 5·43-s + 47-s + 18·49-s + 4·51-s − 6·53-s + 4·57-s − 12·59-s − 3·61-s + 10·63-s − 5·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s − 0.970·17-s − 0.917·19-s + 1.09·21-s − 1.87·23-s − 25-s + 0.962·27-s + 0.371·29-s + 0.359·31-s + 0.870·33-s − 0.986·37-s − 0.160·39-s + 0.937·41-s − 0.762·43-s + 0.145·47-s + 18/7·49-s + 0.560·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.384·61-s + 1.25·63-s − 0.610·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4024\)    =    \(2^{3} \cdot 503\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 4024,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
503 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.75549454920028496858956868410, −6.71365425669421943110451408102, −6.12412770313059561503266987800, −5.79747417983374502173592591723, −4.72217123137914637418913155227, −3.79098191638321667172261619153, −2.91094108943502144802456568326, −2.19139625387954479267748965965, 0, 0, 2.19139625387954479267748965965, 2.91094108943502144802456568326, 3.79098191638321667172261619153, 4.72217123137914637418913155227, 5.79747417983374502173592591723, 6.12412770313059561503266987800, 6.71365425669421943110451408102, 7.75549454920028496858956868410

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