Properties

Degree 2
Conductor $ 2^{4} \cdot 503 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.67·3-s − 2.92·5-s + 1.65·7-s − 0.193·9-s − 5.49·11-s − 4.79·13-s + 4.90·15-s + 5.49·17-s + 0.255·19-s − 2.77·21-s − 3.87·23-s + 3.55·25-s + 5.34·27-s + 2.98·29-s + 5.08·31-s + 9.20·33-s − 4.85·35-s − 0.611·37-s + 8.03·39-s + 3.44·41-s + 3.26·43-s + 0.566·45-s + 2.23·47-s − 4.24·49-s − 9.20·51-s − 1.60·53-s + 16.0·55-s + ⋯
L(s)  = 1  − 0.967·3-s − 1.30·5-s + 0.627·7-s − 0.0645·9-s − 1.65·11-s − 1.32·13-s + 1.26·15-s + 1.33·17-s + 0.0585·19-s − 0.606·21-s − 0.807·23-s + 0.711·25-s + 1.02·27-s + 0.554·29-s + 0.913·31-s + 1.60·33-s − 0.820·35-s − 0.100·37-s + 1.28·39-s + 0.538·41-s + 0.497·43-s + 0.0843·45-s + 0.325·47-s − 0.606·49-s − 1.28·51-s − 0.220·53-s + 2.16·55-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 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 + 2.92T + 5T^{2} \)
7 \( 1 - 1.65T + 7T^{2} \)
11 \( 1 + 5.49T + 11T^{2} \)
13 \( 1 + 4.79T + 13T^{2} \)
17 \( 1 - 5.49T + 17T^{2} \)
19 \( 1 - 0.255T + 19T^{2} \)
23 \( 1 + 3.87T + 23T^{2} \)
29 \( 1 - 2.98T + 29T^{2} \)
31 \( 1 - 5.08T + 31T^{2} \)
37 \( 1 + 0.611T + 37T^{2} \)
41 \( 1 - 3.44T + 41T^{2} \)
43 \( 1 - 3.26T + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + 1.60T + 53T^{2} \)
59 \( 1 - 8.61T + 59T^{2} \)
61 \( 1 + 2.48T + 61T^{2} \)
67 \( 1 - 8.51T + 67T^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 + 0.465T + 73T^{2} \)
79 \( 1 - 0.500T + 79T^{2} \)
83 \( 1 + 8.22T + 83T^{2} \)
89 \( 1 + 0.0767T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.62702589093215239802470859572, −6.93477854908594531005505308712, −5.92395626922714003094735970857, −5.22676459911475690170242635773, −4.89027220609121627224399302703, −4.09025005978182056865156614761, −3.05711722948266556756353058651, −2.36536248603365465805834369093, −0.825386241406238619251554680177, 0, 0.825386241406238619251554680177, 2.36536248603365465805834369093, 3.05711722948266556756353058651, 4.09025005978182056865156614761, 4.89027220609121627224399302703, 5.22676459911475690170242635773, 5.92395626922714003094735970857, 6.93477854908594531005505308712, 7.62702589093215239802470859572

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