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  + 0.596·3-s + 3.74·5-s − 1.10·7-s − 2.64·9-s + 4.05·11-s − 6.28·13-s + 2.23·15-s + 3.42·17-s − 4.55·19-s − 0.661·21-s + 7.12·23-s + 9.00·25-s − 3.36·27-s − 7.58·29-s − 10.1·31-s + 2.41·33-s − 4.14·35-s − 10.9·37-s − 3.74·39-s + 4.41·41-s − 9.01·43-s − 9.89·45-s − 0.789·47-s − 5.77·49-s + 2.04·51-s − 12.5·53-s + 15.1·55-s + ⋯
L(s)  = 1  + 0.344·3-s + 1.67·5-s − 0.418·7-s − 0.881·9-s + 1.22·11-s − 1.74·13-s + 0.576·15-s + 0.831·17-s − 1.04·19-s − 0.144·21-s + 1.48·23-s + 1.80·25-s − 0.648·27-s − 1.40·29-s − 1.82·31-s + 0.421·33-s − 0.701·35-s − 1.79·37-s − 0.600·39-s + 0.688·41-s − 1.37·43-s − 1.47·45-s − 0.115·47-s − 0.824·49-s + 0.286·51-s − 1.72·53-s + 2.04·55-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 \( 1 - 0.596T + 3T^{2} \)
5 \( 1 - 3.74T + 5T^{2} \)
7 \( 1 + 1.10T + 7T^{2} \)
11 \( 1 - 4.05T + 11T^{2} \)
13 \( 1 + 6.28T + 13T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 + 4.55T + 19T^{2} \)
23 \( 1 - 7.12T + 23T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 4.41T + 41T^{2} \)
43 \( 1 + 9.01T + 43T^{2} \)
47 \( 1 + 0.789T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + 3.67T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 8.33T + 67T^{2} \)
71 \( 1 - 16.5T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 1.08T + 79T^{2} \)
83 \( 1 - 8.85T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 7.23T + 97T^{2} \)
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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

−7.32564553852480497886828228566, −6.71157718640696023483080821489, −6.16514703499005645488135288472, −5.25900263195764159535695180866, −5.06464537672769951062121254768, −3.64909689566828681400916470336, −3.04824614649757582429510027301, −2.11522219494197680982942559687, −1.61903122415485114802024391472, 0, 1.61903122415485114802024391472, 2.11522219494197680982942559687, 3.04824614649757582429510027301, 3.64909689566828681400916470336, 5.06464537672769951062121254768, 5.25900263195764159535695180866, 6.16514703499005645488135288472, 6.71157718640696023483080821489, 7.32564553852480497886828228566

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