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  + 2.22·3-s − 3.90·5-s + 0.127·7-s + 1.95·9-s − 1.34·11-s − 1.48·13-s − 8.70·15-s + 1.12·17-s + 3.57·19-s + 0.283·21-s + 6.17·23-s + 10.2·25-s − 2.31·27-s − 0.0940·29-s − 1.35·31-s − 3.00·33-s − 0.497·35-s + 2.38·37-s − 3.31·39-s + 9.91·41-s − 2.37·43-s − 7.65·45-s − 12.8·47-s − 6.98·49-s + 2.50·51-s + 0.100·53-s + 5.27·55-s + ⋯
L(s)  = 1  + 1.28·3-s − 1.74·5-s + 0.0480·7-s + 0.652·9-s − 0.406·11-s − 0.412·13-s − 2.24·15-s + 0.273·17-s + 0.819·19-s + 0.0618·21-s + 1.28·23-s + 2.05·25-s − 0.446·27-s − 0.0174·29-s − 0.243·31-s − 0.522·33-s − 0.0840·35-s + 0.392·37-s − 0.530·39-s + 1.54·41-s − 0.362·43-s − 1.14·45-s − 1.87·47-s − 0.997·49-s + 0.351·51-s + 0.0137·53-s + 0.710·55-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 \( 1 - 2.22T + 3T^{2} \)
5 \( 1 + 3.90T + 5T^{2} \)
7 \( 1 - 0.127T + 7T^{2} \)
11 \( 1 + 1.34T + 11T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 - 3.57T + 19T^{2} \)
23 \( 1 - 6.17T + 23T^{2} \)
29 \( 1 + 0.0940T + 29T^{2} \)
31 \( 1 + 1.35T + 31T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 - 9.91T + 41T^{2} \)
43 \( 1 + 2.37T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 - 0.100T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 8.89T + 61T^{2} \)
67 \( 1 + 5.58T + 67T^{2} \)
71 \( 1 - 9.40T + 71T^{2} \)
73 \( 1 - 4.22T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 0.492T + 83T^{2} \)
89 \( 1 - 8.78T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−7.66472237172724032154042635978, −7.25953395310762713725930406562, −6.26955058780659799920503132091, −5.04737019249926198793155917846, −4.58711067610138190649588562302, −3.60818029003126252730433595338, −3.21514082181748119316281787490, −2.57703599143041058804489271401, −1.26719144578086705595364236449, 0, 1.26719144578086705595364236449, 2.57703599143041058804489271401, 3.21514082181748119316281787490, 3.60818029003126252730433595338, 4.58711067610138190649588562302, 5.04737019249926198793155917846, 6.26955058780659799920503132091, 7.25953395310762713725930406562, 7.66472237172724032154042635978

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