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Properties

Label 2-563-563.562-c0-0-5
Degree $2$
Conductor $563$
Sign $-1$
Analytic cond. $0.280973$
Root an. cond. $0.530069$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

Degree: \(2\)
Conductor: \(563\)
Sign: $-1$
Analytic conductor: \(0.280973\)
Root analytic conductor: \(0.530069\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{563} (562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 563,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6523332469\)
\(L(\frac12)\) \(\approx\) \(0.6523332469\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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 \( 1 + 1.41iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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