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  − 2.89·3-s − 1.96·5-s − 2.82·7-s + 5.39·9-s + 5.91·11-s − 5.24·13-s + 5.68·15-s − 0.885·17-s + 6.74·19-s + 8.19·21-s − 5.73·23-s − 1.14·25-s − 6.93·27-s − 9.96·29-s − 0.549·31-s − 17.1·33-s + 5.55·35-s + 6.03·37-s + 15.1·39-s + 2.81·41-s − 11.4·43-s − 10.5·45-s + 8.25·47-s + 1.00·49-s + 2.56·51-s − 4.66·53-s − 11.6·55-s + ⋯
L(s)  = 1  − 1.67·3-s − 0.878·5-s − 1.06·7-s + 1.79·9-s + 1.78·11-s − 1.45·13-s + 1.46·15-s − 0.214·17-s + 1.54·19-s + 1.78·21-s − 1.19·23-s − 0.228·25-s − 1.33·27-s − 1.84·29-s − 0.0986·31-s − 2.98·33-s + 0.938·35-s + 0.991·37-s + 2.43·39-s + 0.440·41-s − 1.74·43-s − 1.57·45-s + 1.20·47-s + 0.142·49-s + 0.359·51-s − 0.641·53-s − 1.56·55-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 \( 1 + 2.89T + 3T^{2} \)
5 \( 1 + 1.96T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 5.91T + 11T^{2} \)
13 \( 1 + 5.24T + 13T^{2} \)
17 \( 1 + 0.885T + 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
23 \( 1 + 5.73T + 23T^{2} \)
29 \( 1 + 9.96T + 29T^{2} \)
31 \( 1 + 0.549T + 31T^{2} \)
37 \( 1 - 6.03T + 37T^{2} \)
41 \( 1 - 2.81T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 - 8.25T + 47T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 - 1.78T + 59T^{2} \)
61 \( 1 + 7.68T + 61T^{2} \)
67 \( 1 - 2.96T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 - 9.57T + 83T^{2} \)
89 \( 1 - 7.66T + 89T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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

−7.45240583465763744575332202380, −6.54321707737046340266757359642, −6.29712990152613736018760370748, −5.41634316762731115358678167324, −4.77558158172266741068386324813, −3.86832348478206027390444964172, −3.52775349372028174299539159603, −2.03036101382576203136277994762, −0.812381581751239376011451306375, 0, 0.812381581751239376011451306375, 2.03036101382576203136277994762, 3.52775349372028174299539159603, 3.86832348478206027390444964172, 4.77558158172266741068386324813, 5.41634316762731115358678167324, 6.29712990152613736018760370748, 6.54321707737046340266757359642, 7.45240583465763744575332202380

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