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  + 1.72·3-s + 0.586·5-s + 2.46·7-s − 0.0259·9-s − 2.73·11-s + 1.16·13-s + 1.01·15-s − 3.49·17-s − 5.19·19-s + 4.24·21-s + 1.89·23-s − 4.65·25-s − 5.21·27-s − 3.57·29-s + 1.26·31-s − 4.70·33-s + 1.44·35-s + 10.5·37-s + 2.00·39-s − 4.19·41-s − 8.57·43-s − 0.0152·45-s − 4.29·47-s − 0.944·49-s − 6.02·51-s − 6.91·53-s − 1.60·55-s + ⋯
L(s)  = 1  + 0.995·3-s + 0.262·5-s + 0.930·7-s − 0.00864·9-s − 0.823·11-s + 0.322·13-s + 0.261·15-s − 0.847·17-s − 1.19·19-s + 0.926·21-s + 0.395·23-s − 0.931·25-s − 1.00·27-s − 0.663·29-s + 0.226·31-s − 0.819·33-s + 0.243·35-s + 1.73·37-s + 0.321·39-s − 0.654·41-s − 1.30·43-s − 0.00226·45-s − 0.626·47-s − 0.134·49-s − 0.843·51-s − 0.949·53-s − 0.215·55-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 \( 1 - 1.72T + 3T^{2} \)
5 \( 1 - 0.586T + 5T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 - 1.16T + 13T^{2} \)
17 \( 1 + 3.49T + 17T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 + 3.57T + 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 4.19T + 41T^{2} \)
43 \( 1 + 8.57T + 43T^{2} \)
47 \( 1 + 4.29T + 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 - 6.43T + 59T^{2} \)
61 \( 1 - 5.28T + 61T^{2} \)
67 \( 1 + 7.31T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 3.03T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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

−7.80423649762654864631379822950, −6.86819450909751376193711337125, −6.10680975867818024054844832011, −5.34046125467184399400983144084, −4.58495749687462894153920863215, −3.89333281101821933957842380270, −2.94150766501646533810362417266, −2.23310393507246221682490563436, −1.63554498916254410383738748312, 0, 1.63554498916254410383738748312, 2.23310393507246221682490563436, 2.94150766501646533810362417266, 3.89333281101821933957842380270, 4.58495749687462894153920863215, 5.34046125467184399400983144084, 6.10680975867818024054844832011, 6.86819450909751376193711337125, 7.80423649762654864631379822950

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