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  + 1.51·3-s − 2.23·5-s + 3.60·7-s − 0.717·9-s + 1.50·11-s + 0.00459·13-s − 3.37·15-s + 1.11·17-s − 2.49·19-s + 5.44·21-s − 1.72·23-s − 0.00724·25-s − 5.61·27-s − 0.572·29-s − 3.25·31-s + 2.27·33-s − 8.05·35-s − 6.13·37-s + 0.00694·39-s − 5.89·41-s + 7.66·43-s + 1.60·45-s − 6.44·47-s + 5.98·49-s + 1.69·51-s − 7.82·53-s − 3.35·55-s + ⋯
L(s)  = 1  + 0.872·3-s − 0.999·5-s + 1.36·7-s − 0.239·9-s + 0.453·11-s + 0.00127·13-s − 0.871·15-s + 0.271·17-s − 0.573·19-s + 1.18·21-s − 0.359·23-s − 0.00144·25-s − 1.08·27-s − 0.106·29-s − 0.584·31-s + 0.395·33-s − 1.36·35-s − 1.00·37-s + 0.00111·39-s − 0.920·41-s + 1.16·43-s + 0.239·45-s − 0.940·47-s + 0.854·49-s + 0.236·51-s − 1.07·53-s − 0.452·55-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 \( 1 - 1.51T + 3T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
7 \( 1 - 3.60T + 7T^{2} \)
11 \( 1 - 1.50T + 11T^{2} \)
13 \( 1 - 0.00459T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 + 2.49T + 19T^{2} \)
23 \( 1 + 1.72T + 23T^{2} \)
29 \( 1 + 0.572T + 29T^{2} \)
31 \( 1 + 3.25T + 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 + 5.89T + 41T^{2} \)
43 \( 1 - 7.66T + 43T^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 + 7.82T + 53T^{2} \)
59 \( 1 + 0.253T + 59T^{2} \)
61 \( 1 - 7.08T + 61T^{2} \)
67 \( 1 + 6.16T + 67T^{2} \)
71 \( 1 - 9.32T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 1.99T + 79T^{2} \)
83 \( 1 + 6.91T + 83T^{2} \)
89 \( 1 - 3.37T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−7.65393421339527115881028391813, −7.11592885624991137059105681963, −6.08991868353795356816579626458, −5.26520357004341806976422661177, −4.49750155281297289629946743513, −3.84897562994749473276149429719, −3.23232395641563772444107442294, −2.18338612849222785981301334443, −1.47456512985157684613966121565, 0, 1.47456512985157684613966121565, 2.18338612849222785981301334443, 3.23232395641563772444107442294, 3.84897562994749473276149429719, 4.49750155281297289629946743513, 5.26520357004341806976422661177, 6.08991868353795356816579626458, 7.11592885624991137059105681963, 7.65393421339527115881028391813

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