Properties

Label 1-563-563.562-r1-0-0
Degree $1$
Conductor $563$
Sign $1$
Analytic cond. $60.5027$
Root an. cond. $60.5027$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(563\)
Sign: $1$
Analytic conductor: \(60.5027\)
Root analytic conductor: \(60.5027\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{563} (562, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 563,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.381121446\)
\(L(\frac12)\) \(\approx\) \(2.381121446\)
\(L(1)\) \(\approx\) \(1.191621100\)
\(L(1)\) \(\approx\) \(1.191621100\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad563 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.44398396056945250483541858602, −22.04688056142865267272281319893, −20.85236240398872433793148304114, −20.49367029895660009637199282704, −19.753442804007359156945374507126, −18.7179395741160821605018920845, −18.5356527640599964066818025206, −17.21224114116395438894820804909, −16.34269453977467928879784966058, −15.47757706224107021636988662965, −14.78364876645570558289215952034, −14.07998120294258143747645305513, −12.65908168737957057819821146728, −11.640812195304594641934990461882, −11.1015026366101615365813018130, −9.93214053653488081166099255384, −8.86344094072557433132551048165, −8.43850349534664769136122502358, −7.49723553068968270930933155930, −6.95543557580235650221803021336, −5.326258992365120469678467119688, −3.81549781393358963936410474048, −3.246419659457123454784826618553, −1.687605946571056968139985934980, −1.00820339867981708478766173874, 1.00820339867981708478766173874, 1.687605946571056968139985934980, 3.246419659457123454784826618553, 3.81549781393358963936410474048, 5.326258992365120469678467119688, 6.95543557580235650221803021336, 7.49723553068968270930933155930, 8.43850349534664769136122502358, 8.86344094072557433132551048165, 9.93214053653488081166099255384, 11.1015026366101615365813018130, 11.640812195304594641934990461882, 12.65908168737957057819821146728, 14.07998120294258143747645305513, 14.78364876645570558289215952034, 15.47757706224107021636988662965, 16.34269453977467928879784966058, 17.21224114116395438894820804909, 18.5356527640599964066818025206, 18.7179395741160821605018920845, 19.753442804007359156945374507126, 20.49367029895660009637199282704, 20.85236240398872433793148304114, 22.04688056142865267272281319893, 23.44398396056945250483541858602

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