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  − 0.950·3-s − 2.28·5-s − 2.71·7-s − 2.09·9-s − 1.36·11-s − 2.93·13-s + 2.17·15-s − 2.61·17-s + 7.79·19-s + 2.57·21-s + 2.61·23-s + 0.230·25-s + 4.84·27-s − 0.314·29-s + 7.95·31-s + 1.30·33-s + 6.19·35-s − 4.17·37-s + 2.78·39-s + 6.16·41-s − 0.457·43-s + 4.79·45-s − 7.67·47-s + 0.345·49-s + 2.48·51-s + 7.26·53-s + 3.13·55-s + ⋯
L(s)  = 1  − 0.548·3-s − 1.02·5-s − 1.02·7-s − 0.699·9-s − 0.412·11-s − 0.812·13-s + 0.561·15-s − 0.633·17-s + 1.78·19-s + 0.561·21-s + 0.544·23-s + 0.0460·25-s + 0.932·27-s − 0.0583·29-s + 1.42·31-s + 0.226·33-s + 1.04·35-s − 0.686·37-s + 0.445·39-s + 0.962·41-s − 0.0698·43-s + 0.715·45-s − 1.11·47-s + 0.0493·49-s + 0.347·51-s + 0.997·53-s + 0.422·55-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 \( 1 + 0.950T + 3T^{2} \)
5 \( 1 + 2.28T + 5T^{2} \)
7 \( 1 + 2.71T + 7T^{2} \)
11 \( 1 + 1.36T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + 2.61T + 17T^{2} \)
19 \( 1 - 7.79T + 19T^{2} \)
23 \( 1 - 2.61T + 23T^{2} \)
29 \( 1 + 0.314T + 29T^{2} \)
31 \( 1 - 7.95T + 31T^{2} \)
37 \( 1 + 4.17T + 37T^{2} \)
41 \( 1 - 6.16T + 41T^{2} \)
43 \( 1 + 0.457T + 43T^{2} \)
47 \( 1 + 7.67T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 + 0.217T + 59T^{2} \)
61 \( 1 - 7.26T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 16.5T + 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 - 16.0T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 0.0952T + 89T^{2} \)
97 \( 1 + 9.27T + 97T^{2} \)
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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

−7.44335898013687975263001788768, −6.80635525327280845894496646744, −6.17139431624754729187976847829, −5.27676297461432356945986576816, −4.84521580484567033565388552192, −3.81113690977989338372067634145, −3.11322080480123855014306698035, −2.51012258927727108144791064871, −0.831289083494322303812140328840, 0, 0.831289083494322303812140328840, 2.51012258927727108144791064871, 3.11322080480123855014306698035, 3.81113690977989338372067634145, 4.84521580484567033565388552192, 5.27676297461432356945986576816, 6.17139431624754729187976847829, 6.80635525327280845894496646744, 7.44335898013687975263001788768

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