Properties

Degree $2$
Conductor $574$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 5-s − 3·6-s + 7-s + 8-s + 6·9-s − 10-s − 2·11-s − 3·12-s + 14-s + 3·15-s + 16-s − 3·17-s + 6·18-s − 8·19-s − 20-s − 3·21-s − 2·22-s − 4·23-s − 3·24-s − 4·25-s − 9·27-s + 28-s − 5·29-s + 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.603·11-s − 0.866·12-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 1.83·19-s − 0.223·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.612·24-s − 4/5·25-s − 1.73·27-s + 0.188·28-s − 0.928·29-s + 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(574\)    =    \(2 \cdot 7 \cdot 41\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{574} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 574,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
41 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−10.82348754711184792755959334111, −9.814547451875919314932987373145, −8.260721856204501813182654193846, −7.29964473624905066637087401045, −6.31811319083545584980505157205, −5.70227872805281049394495893274, −4.65757518643900400132339268063, −4.04251778571385567301847619669, −2.03110414024071717045287920222, 0, 2.03110414024071717045287920222, 4.04251778571385567301847619669, 4.65757518643900400132339268063, 5.70227872805281049394495893274, 6.31811319083545584980505157205, 7.29964473624905066637087401045, 8.260721856204501813182654193846, 9.814547451875919314932987373145, 10.82348754711184792755959334111

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