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  + 3.02·3-s − 1.92·5-s − 4.18·7-s + 6.16·9-s + 1.99·11-s + 1.74·13-s − 5.82·15-s − 2.18·17-s + 1.78·19-s − 12.6·21-s + 3.66·23-s − 1.29·25-s + 9.58·27-s − 4.48·29-s − 10.3·31-s + 6.05·33-s + 8.05·35-s + 1.81·37-s + 5.28·39-s − 5.92·41-s − 3.30·43-s − 11.8·45-s − 0.751·47-s + 10.5·49-s − 6.60·51-s − 1.29·53-s − 3.84·55-s + ⋯
L(s)  = 1  + 1.74·3-s − 0.860·5-s − 1.58·7-s + 2.05·9-s + 0.602·11-s + 0.484·13-s − 1.50·15-s − 0.529·17-s + 0.408·19-s − 2.76·21-s + 0.764·23-s − 0.259·25-s + 1.84·27-s − 0.832·29-s − 1.85·31-s + 1.05·33-s + 1.36·35-s + 0.297·37-s + 0.846·39-s − 0.925·41-s − 0.503·43-s − 1.76·45-s − 0.109·47-s + 1.50·49-s − 0.925·51-s − 0.178·53-s − 0.518·55-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 \( 1 - 3.02T + 3T^{2} \)
5 \( 1 + 1.92T + 5T^{2} \)
7 \( 1 + 4.18T + 7T^{2} \)
11 \( 1 - 1.99T + 11T^{2} \)
13 \( 1 - 1.74T + 13T^{2} \)
17 \( 1 + 2.18T + 17T^{2} \)
19 \( 1 - 1.78T + 19T^{2} \)
23 \( 1 - 3.66T + 23T^{2} \)
29 \( 1 + 4.48T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 1.81T + 37T^{2} \)
41 \( 1 + 5.92T + 41T^{2} \)
43 \( 1 + 3.30T + 43T^{2} \)
47 \( 1 + 0.751T + 47T^{2} \)
53 \( 1 + 1.29T + 53T^{2} \)
59 \( 1 - 2.89T + 59T^{2} \)
61 \( 1 + 0.765T + 61T^{2} \)
67 \( 1 - 0.202T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 0.211T + 79T^{2} \)
83 \( 1 - 9.33T + 83T^{2} \)
89 \( 1 - 0.912T + 89T^{2} \)
97 \( 1 + 7.66T + 97T^{2} \)
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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

−7.54632393284370257440016124982, −6.96511070946395732217334632217, −6.44175608069209232679497563708, −5.34599345068758358431213199958, −4.14452805745429540690567255855, −3.63686273515912182316328744674, −3.33527488697075737048864264247, −2.47140217141521147729406677066, −1.46436350369554941712900149375, 0, 1.46436350369554941712900149375, 2.47140217141521147729406677066, 3.33527488697075737048864264247, 3.63686273515912182316328744674, 4.14452805745429540690567255855, 5.34599345068758358431213199958, 6.44175608069209232679497563708, 6.96511070946395732217334632217, 7.54632393284370257440016124982

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