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  + 2.92·3-s + 0.124·5-s − 2.19·7-s + 5.56·9-s − 0.684·11-s − 5.23·13-s + 0.365·15-s + 3.19·17-s + 1.04·19-s − 6.43·21-s − 6.34·23-s − 4.98·25-s + 7.50·27-s − 6.90·29-s + 1.53·31-s − 2.00·33-s − 0.274·35-s − 1.84·37-s − 15.3·39-s + 10.2·41-s + 6.72·43-s + 0.695·45-s + 2.89·47-s − 2.16·49-s + 9.36·51-s − 6.45·53-s − 0.0855·55-s + ⋯
L(s)  = 1  + 1.68·3-s + 0.0558·5-s − 0.830·7-s + 1.85·9-s − 0.206·11-s − 1.45·13-s + 0.0944·15-s + 0.776·17-s + 0.238·19-s − 1.40·21-s − 1.32·23-s − 0.996·25-s + 1.44·27-s − 1.28·29-s + 0.275·31-s − 0.348·33-s − 0.0464·35-s − 0.303·37-s − 2.45·39-s + 1.59·41-s + 1.02·43-s + 0.103·45-s + 0.421·47-s − 0.309·49-s + 1.31·51-s − 0.886·53-s − 0.0115·55-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 \( 1 - 2.92T + 3T^{2} \)
5 \( 1 - 0.124T + 5T^{2} \)
7 \( 1 + 2.19T + 7T^{2} \)
11 \( 1 + 0.684T + 11T^{2} \)
13 \( 1 + 5.23T + 13T^{2} \)
17 \( 1 - 3.19T + 17T^{2} \)
19 \( 1 - 1.04T + 19T^{2} \)
23 \( 1 + 6.34T + 23T^{2} \)
29 \( 1 + 6.90T + 29T^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + 1.84T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 - 2.89T + 47T^{2} \)
53 \( 1 + 6.45T + 53T^{2} \)
59 \( 1 + 3.81T + 59T^{2} \)
61 \( 1 - 9.92T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 - 2.63T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 - 0.990T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 1.30T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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

−7.63783434196779930286495777420, −7.17542377265190635646503254721, −6.11286026300826016041461376136, −5.44810709553666830548893361619, −4.29559740669546450199408610890, −3.81923878047502442306845545522, −2.93265522716764507611608850374, −2.47532460984268931622867493895, −1.61824800684637868941273546609, 0, 1.61824800684637868941273546609, 2.47532460984268931622867493895, 2.93265522716764507611608850374, 3.81923878047502442306845545522, 4.29559740669546450199408610890, 5.44810709553666830548893361619, 6.11286026300826016041461376136, 7.17542377265190635646503254721, 7.63783434196779930286495777420

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