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  − 2.84·3-s − 2.72·5-s + 3.71·7-s + 5.09·9-s + 3.83·11-s + 2.85·13-s + 7.74·15-s + 8.15·17-s − 3.41·19-s − 10.5·21-s − 0.915·23-s + 2.40·25-s − 5.96·27-s − 9.16·29-s − 5.69·31-s − 10.9·33-s − 10.1·35-s − 8.85·37-s − 8.11·39-s − 10.6·41-s + 11.1·43-s − 13.8·45-s + 3.44·47-s + 6.81·49-s − 23.2·51-s − 4.95·53-s − 10.4·55-s + ⋯
L(s)  = 1  − 1.64·3-s − 1.21·5-s + 1.40·7-s + 1.69·9-s + 1.15·11-s + 0.790·13-s + 1.99·15-s + 1.97·17-s − 0.783·19-s − 2.30·21-s − 0.190·23-s + 0.480·25-s − 1.14·27-s − 1.70·29-s − 1.02·31-s − 1.90·33-s − 1.70·35-s − 1.45·37-s − 1.29·39-s − 1.66·41-s + 1.70·43-s − 2.06·45-s + 0.502·47-s + 0.973·49-s − 3.24·51-s − 0.680·53-s − 1.40·55-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 \( 1 + 2.84T + 3T^{2} \)
5 \( 1 + 2.72T + 5T^{2} \)
7 \( 1 - 3.71T + 7T^{2} \)
11 \( 1 - 3.83T + 11T^{2} \)
13 \( 1 - 2.85T + 13T^{2} \)
17 \( 1 - 8.15T + 17T^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
23 \( 1 + 0.915T + 23T^{2} \)
29 \( 1 + 9.16T + 29T^{2} \)
31 \( 1 + 5.69T + 31T^{2} \)
37 \( 1 + 8.85T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 3.44T + 47T^{2} \)
53 \( 1 + 4.95T + 53T^{2} \)
59 \( 1 + 2.81T + 59T^{2} \)
61 \( 1 + 3.59T + 61T^{2} \)
67 \( 1 - 9.01T + 67T^{2} \)
71 \( 1 - 7.15T + 71T^{2} \)
73 \( 1 + 7.82T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 6.47T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−7.46603568532477633752989114507, −6.81316368566855053356494598013, −5.92548965263457136085193900077, −5.43079686590185439650330899877, −4.78406032447302733496722165849, −3.83674891249201893422897380886, −3.70681285271605684468007757782, −1.69767091313317834110523203051, −1.16455713802300826113658606586, 0, 1.16455713802300826113658606586, 1.69767091313317834110523203051, 3.70681285271605684468007757782, 3.83674891249201893422897380886, 4.78406032447302733496722165849, 5.43079686590185439650330899877, 5.92548965263457136085193900077, 6.81316368566855053356494598013, 7.46603568532477633752989114507

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