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  + 2.40·3-s + 0.590·5-s − 1.95·7-s + 2.78·9-s + 1.52·11-s − 4.67·13-s + 1.41·15-s − 3.04·17-s − 0.338·19-s − 4.71·21-s + 7.98·23-s − 4.65·25-s − 0.513·27-s + 6.01·29-s + 4.17·31-s + 3.67·33-s − 1.15·35-s − 11.0·37-s − 11.2·39-s − 4.82·41-s − 12.4·43-s + 1.64·45-s − 11.0·47-s − 3.16·49-s − 7.33·51-s − 3.43·53-s + 0.901·55-s + ⋯
L(s)  = 1  + 1.38·3-s + 0.263·5-s − 0.740·7-s + 0.928·9-s + 0.460·11-s − 1.29·13-s + 0.366·15-s − 0.739·17-s − 0.0775·19-s − 1.02·21-s + 1.66·23-s − 0.930·25-s − 0.0988·27-s + 1.11·29-s + 0.749·31-s + 0.639·33-s − 0.195·35-s − 1.81·37-s − 1.80·39-s − 0.753·41-s − 1.89·43-s + 0.245·45-s − 1.60·47-s − 0.451·49-s − 1.02·51-s − 0.472·53-s + 0.121·55-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 \( 1 - 2.40T + 3T^{2} \)
5 \( 1 - 0.590T + 5T^{2} \)
7 \( 1 + 1.95T + 7T^{2} \)
11 \( 1 - 1.52T + 11T^{2} \)
13 \( 1 + 4.67T + 13T^{2} \)
17 \( 1 + 3.04T + 17T^{2} \)
19 \( 1 + 0.338T + 19T^{2} \)
23 \( 1 - 7.98T + 23T^{2} \)
29 \( 1 - 6.01T + 29T^{2} \)
31 \( 1 - 4.17T + 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 + 2.63T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + 9.66T + 71T^{2} \)
73 \( 1 + 6.33T + 73T^{2} \)
79 \( 1 - 0.667T + 79T^{2} \)
83 \( 1 - 2.26T + 83T^{2} \)
89 \( 1 - 2.21T + 89T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−7.49722904375810744840002044630, −6.77542056464147596411211759338, −6.48266444309668303799997857326, −5.15651086161379366626457229209, −4.68240091146949948276119229558, −3.55822240707759241407342831285, −3.12247533371052104727667212527, −2.36857156174397122903587858360, −1.58510729611366036278814425202, 0, 1.58510729611366036278814425202, 2.36857156174397122903587858360, 3.12247533371052104727667212527, 3.55822240707759241407342831285, 4.68240091146949948276119229558, 5.15651086161379366626457229209, 6.48266444309668303799997857326, 6.77542056464147596411211759338, 7.49722904375810744840002044630

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