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  + 1.85·3-s + 1.44·5-s + 1.96·7-s + 0.453·9-s − 2.85·11-s − 3.84·13-s + 2.68·15-s + 1.30·17-s − 3.53·19-s + 3.65·21-s − 4.20·23-s − 2.91·25-s − 4.73·27-s − 1.10·29-s + 3.53·31-s − 5.30·33-s + 2.83·35-s − 6.61·37-s − 7.14·39-s + 0.671·41-s + 4.32·43-s + 0.653·45-s + 0.378·47-s − 3.13·49-s + 2.42·51-s − 7.13·53-s − 4.12·55-s + ⋯
L(s)  = 1  + 1.07·3-s + 0.645·5-s + 0.742·7-s + 0.151·9-s − 0.861·11-s − 1.06·13-s + 0.692·15-s + 0.316·17-s − 0.810·19-s + 0.796·21-s − 0.877·23-s − 0.583·25-s − 0.910·27-s − 0.205·29-s + 0.634·31-s − 0.923·33-s + 0.479·35-s − 1.08·37-s − 1.14·39-s + 0.104·41-s + 0.659·43-s + 0.0974·45-s + 0.0552·47-s − 0.448·49-s + 0.339·51-s − 0.980·53-s − 0.555·55-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 \( 1 - 1.85T + 3T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 - 1.96T + 7T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 + 3.53T + 19T^{2} \)
23 \( 1 + 4.20T + 23T^{2} \)
29 \( 1 + 1.10T + 29T^{2} \)
31 \( 1 - 3.53T + 31T^{2} \)
37 \( 1 + 6.61T + 37T^{2} \)
41 \( 1 - 0.671T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 - 0.378T + 47T^{2} \)
53 \( 1 + 7.13T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 6.51T + 61T^{2} \)
67 \( 1 - 7.55T + 67T^{2} \)
71 \( 1 + 0.0744T + 71T^{2} \)
73 \( 1 - 3.29T + 73T^{2} \)
79 \( 1 + 8.57T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 + 8.01T + 97T^{2} \)
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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

−7.66488720114680914347011068776, −7.00723828939002764931063882298, −5.96697715185238878348302423427, −5.40818575969720963358227364996, −4.64534635112977684771867286561, −3.85315372719958405065143697022, −2.84943288730317361237745264838, −2.28081791294447833927469708487, −1.68386886340034345648761537884, 0, 1.68386886340034345648761537884, 2.28081791294447833927469708487, 2.84943288730317361237745264838, 3.85315372719958405065143697022, 4.64534635112977684771867286561, 5.40818575969720963358227364996, 5.96697715185238878348302423427, 7.00723828939002764931063882298, 7.66488720114680914347011068776

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