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  + 0.893·3-s − 3.85·5-s − 3.77·7-s − 2.20·9-s + 4.71·11-s − 0.511·13-s − 3.44·15-s + 5.63·17-s − 0.0695·19-s − 3.37·21-s − 5.20·23-s + 9.89·25-s − 4.64·27-s − 1.66·29-s + 0.370·31-s + 4.21·33-s + 14.5·35-s + 0.932·37-s − 0.457·39-s + 5.06·41-s + 6.38·43-s + 8.49·45-s + 4.19·47-s + 7.26·49-s + 5.03·51-s + 1.31·53-s − 18.2·55-s + ⋯
L(s)  = 1  + 0.516·3-s − 1.72·5-s − 1.42·7-s − 0.733·9-s + 1.42·11-s − 0.141·13-s − 0.890·15-s + 1.36·17-s − 0.0159·19-s − 0.736·21-s − 1.08·23-s + 1.97·25-s − 0.894·27-s − 0.310·29-s + 0.0665·31-s + 0.734·33-s + 2.46·35-s + 0.153·37-s − 0.0732·39-s + 0.790·41-s + 0.974·43-s + 1.26·45-s + 0.612·47-s + 1.03·49-s + 0.704·51-s + 0.179·53-s − 2.45·55-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 \( 1 - 0.893T + 3T^{2} \)
5 \( 1 + 3.85T + 5T^{2} \)
7 \( 1 + 3.77T + 7T^{2} \)
11 \( 1 - 4.71T + 11T^{2} \)
13 \( 1 + 0.511T + 13T^{2} \)
17 \( 1 - 5.63T + 17T^{2} \)
19 \( 1 + 0.0695T + 19T^{2} \)
23 \( 1 + 5.20T + 23T^{2} \)
29 \( 1 + 1.66T + 29T^{2} \)
31 \( 1 - 0.370T + 31T^{2} \)
37 \( 1 - 0.932T + 37T^{2} \)
41 \( 1 - 5.06T + 41T^{2} \)
43 \( 1 - 6.38T + 43T^{2} \)
47 \( 1 - 4.19T + 47T^{2} \)
53 \( 1 - 1.31T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 2.09T + 67T^{2} \)
71 \( 1 + 3.30T + 71T^{2} \)
73 \( 1 - 1.07T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−7.39694754406725891929023584889, −7.05422610081746680270074972831, −6.11687428484342205946077659409, −5.56972624544941969683505610023, −4.13718574126283258647986959944, −3.89273730827502349557882089362, −3.27044749342172979787537287447, −2.57252893128821520769501118820, −1.00057532283843358102770025283, 0, 1.00057532283843358102770025283, 2.57252893128821520769501118820, 3.27044749342172979787537287447, 3.89273730827502349557882089362, 4.13718574126283258647986959944, 5.56972624544941969683505610023, 6.11687428484342205946077659409, 7.05422610081746680270074972831, 7.39694754406725891929023584889

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