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  − 2.11·3-s + 2.26·5-s + 4.06·7-s + 1.46·9-s + 1.00·11-s − 5.09·13-s − 4.78·15-s + 1.45·17-s − 6.53·19-s − 8.58·21-s + 4.54·23-s + 0.136·25-s + 3.25·27-s − 6.23·29-s + 10.1·31-s − 2.11·33-s + 9.21·35-s + 3.44·37-s + 10.7·39-s − 9.13·41-s − 2.78·43-s + 3.31·45-s − 6.12·47-s + 9.52·49-s − 3.07·51-s − 3.12·53-s + 2.27·55-s + ⋯
L(s)  = 1  − 1.21·3-s + 1.01·5-s + 1.53·7-s + 0.486·9-s + 0.302·11-s − 1.41·13-s − 1.23·15-s + 0.352·17-s − 1.50·19-s − 1.87·21-s + 0.948·23-s + 0.0272·25-s + 0.625·27-s − 1.15·29-s + 1.81·31-s − 0.368·33-s + 1.55·35-s + 0.566·37-s + 1.72·39-s − 1.42·41-s − 0.424·43-s + 0.493·45-s − 0.892·47-s + 1.36·49-s − 0.430·51-s − 0.428·53-s + 0.306·55-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 \( 1 + 2.11T + 3T^{2} \)
5 \( 1 - 2.26T + 5T^{2} \)
7 \( 1 - 4.06T + 7T^{2} \)
11 \( 1 - 1.00T + 11T^{2} \)
13 \( 1 + 5.09T + 13T^{2} \)
17 \( 1 - 1.45T + 17T^{2} \)
19 \( 1 + 6.53T + 19T^{2} \)
23 \( 1 - 4.54T + 23T^{2} \)
29 \( 1 + 6.23T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 - 3.44T + 37T^{2} \)
41 \( 1 + 9.13T + 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 + 6.12T + 47T^{2} \)
53 \( 1 + 3.12T + 53T^{2} \)
59 \( 1 + 2.23T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 0.606T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 6.43T + 83T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−7.44039906390074327001600447908, −6.52483692103614657570441647452, −6.12975627675524119014618592660, −5.26139433234333713970792776152, −4.87138390840754058186233181400, −4.38406021332297631995194387380, −2.89599594291914374386733693364, −1.97385220362694397831152403261, −1.34035791832109570084665018600, 0, 1.34035791832109570084665018600, 1.97385220362694397831152403261, 2.89599594291914374386733693364, 4.38406021332297631995194387380, 4.87138390840754058186233181400, 5.26139433234333713970792776152, 6.12975627675524119014618592660, 6.52483692103614657570441647452, 7.44039906390074327001600447908

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