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  + 1.98·3-s + 0.688·5-s + 1.63·7-s + 0.950·9-s − 1.13·11-s − 4.69·13-s + 1.36·15-s + 3.66·17-s + 0.478·19-s + 3.25·21-s − 4.93·23-s − 4.52·25-s − 4.07·27-s + 3.04·29-s − 7.78·31-s − 2.25·33-s + 1.12·35-s − 1.05·37-s − 9.32·39-s − 10.5·41-s + 3.06·43-s + 0.654·45-s − 11.5·47-s − 4.32·49-s + 7.28·51-s + 6.59·53-s − 0.781·55-s + ⋯
L(s)  = 1  + 1.14·3-s + 0.307·5-s + 0.618·7-s + 0.316·9-s − 0.342·11-s − 1.30·13-s + 0.353·15-s + 0.888·17-s + 0.109·19-s + 0.709·21-s − 1.02·23-s − 0.905·25-s − 0.783·27-s + 0.565·29-s − 1.39·31-s − 0.392·33-s + 0.190·35-s − 0.174·37-s − 1.49·39-s − 1.65·41-s + 0.468·43-s + 0.0976·45-s − 1.68·47-s − 0.617·49-s + 1.01·51-s + 0.905·53-s − 0.105·55-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 \( 1 - 1.98T + 3T^{2} \)
5 \( 1 - 0.688T + 5T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 + 1.13T + 11T^{2} \)
13 \( 1 + 4.69T + 13T^{2} \)
17 \( 1 - 3.66T + 17T^{2} \)
19 \( 1 - 0.478T + 19T^{2} \)
23 \( 1 + 4.93T + 23T^{2} \)
29 \( 1 - 3.04T + 29T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 + 1.05T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 3.06T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 6.59T + 53T^{2} \)
59 \( 1 - 2.97T + 59T^{2} \)
61 \( 1 + 9.43T + 61T^{2} \)
67 \( 1 - 8.15T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 2.52T + 73T^{2} \)
79 \( 1 - 0.930T + 79T^{2} \)
83 \( 1 + 1.69T + 83T^{2} \)
89 \( 1 - 1.82T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−7.71186497925010243460438877854, −7.07210472980103811211983505436, −6.01540342546925623228279012825, −5.32434737095783305198806645257, −4.68892818634659626201777614856, −3.69809676168292337458488503030, −3.07757290967532801326257403068, −2.16791010395709344372487601855, −1.69422163749019850652635472447, 0, 1.69422163749019850652635472447, 2.16791010395709344372487601855, 3.07757290967532801326257403068, 3.69809676168292337458488503030, 4.68892818634659626201777614856, 5.32434737095783305198806645257, 6.01540342546925623228279012825, 7.07210472980103811211983505436, 7.71186497925010243460438877854

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