Properties

Degree 2
Conductor 563
Sign $-1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·5-s − 1.41i·6-s + 7-s − 2.00·10-s + 1.00·12-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 17-s + 19-s − 1.41i·20-s − 21-s + ⋯
L(s)  = 1  + 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·5-s − 1.41i·6-s + 7-s − 2.00·10-s + 1.00·12-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 17-s + 19-s − 1.41i·20-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(563\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(0\)
character  :  $\chi_{563} (562, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 563,\ (\ :0),\ -1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6523332469\)
\(L(\frac12)\)  \(\approx\)  \(0.6523332469\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 563$,\(F_p(T)\) is a polynomial of degree 2. If $p = 563$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad563 \( 1 + T \)
good2 \( 1 - 1.41iT - T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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

−11.28187127132090665349358956268, −10.81729902141978609999300877975, −9.620196492387458257340825716854, −8.331956355553905437178219978167, −7.58954304067215796392683223779, −6.78642278381194156730978081496, −6.07782034797219569215891263814, −5.36190289739853469999498574010, −4.28127219713578690712690574835, −2.47192466332174092058953421171, 0.925630044959445458646108862088, 2.09780978497663742922633009838, 3.86133284679924835613565817504, 4.89426354768251895145989938166, 5.37499563720585411900221105330, 6.81321085057391180508905555377, 8.283771820274635487078762950557, 8.923553383035821704262623852138, 9.935318588558455993266471417697, 10.83395599285513975218397484582

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