Properties

Degree 2
Conductor $ 2^{4} \cdot 503 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.447·3-s + 0.901·5-s + 1.57·7-s − 2.79·9-s − 2.73·11-s − 0.127·13-s + 0.403·15-s − 0.965·17-s − 1.08·19-s + 0.706·21-s − 0.734·23-s − 4.18·25-s − 2.59·27-s + 9.81·29-s + 6.56·31-s − 1.22·33-s + 1.42·35-s − 2.91·37-s − 0.0572·39-s + 1.43·41-s + 2.61·43-s − 2.52·45-s − 2.70·47-s − 4.50·49-s − 0.432·51-s + 0.901·53-s − 2.46·55-s + ⋯
L(s)  = 1  + 0.258·3-s + 0.403·5-s + 0.596·7-s − 0.933·9-s − 0.824·11-s − 0.0354·13-s + 0.104·15-s − 0.234·17-s − 0.248·19-s + 0.154·21-s − 0.153·23-s − 0.837·25-s − 0.499·27-s + 1.82·29-s + 1.17·31-s − 0.213·33-s + 0.240·35-s − 0.479·37-s − 0.00916·39-s + 0.223·41-s + 0.398·43-s − 0.376·45-s − 0.394·47-s − 0.643·49-s − 0.0604·51-s + 0.123·53-s − 0.332·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8048\)    =    \(2^{4} \cdot 503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8048} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8048,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 0.447T + 3T^{2} \)
5 \( 1 - 0.901T + 5T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 + 0.127T + 13T^{2} \)
17 \( 1 + 0.965T + 17T^{2} \)
19 \( 1 + 1.08T + 19T^{2} \)
23 \( 1 + 0.734T + 23T^{2} \)
29 \( 1 - 9.81T + 29T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + 2.91T + 37T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 - 2.61T + 43T^{2} \)
47 \( 1 + 2.70T + 47T^{2} \)
53 \( 1 - 0.901T + 53T^{2} \)
59 \( 1 + 6.31T + 59T^{2} \)
61 \( 1 + 5.52T + 61T^{2} \)
67 \( 1 - 3.88T + 67T^{2} \)
71 \( 1 - 9.68T + 71T^{2} \)
73 \( 1 + 1.43T + 73T^{2} \)
79 \( 1 - 9.80T + 79T^{2} \)
83 \( 1 + 4.88T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 8.63T + 97T^{2} \)
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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

−7.76119871012325711760405027311, −6.66183052510014629699206097963, −6.13934469778991594315738722988, −5.31569306188210573666778152668, −4.80461035753557373998870521082, −3.91505954206529786726157457458, −2.82193387793391494935573333561, −2.43569672532271700121798305163, −1.35105329859266142473500319013, 0, 1.35105329859266142473500319013, 2.43569672532271700121798305163, 2.82193387793391494935573333561, 3.91505954206529786726157457458, 4.80461035753557373998870521082, 5.31569306188210573666778152668, 6.13934469778991594315738722988, 6.66183052510014629699206097963, 7.76119871012325711760405027311

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