Properties

Degree $2$
Conductor $5610$
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-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 11-s + 12-s + 5·13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 5·19-s + 20-s − 21-s − 22-s − 5·23-s − 24-s + 25-s − 5·26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.218·21-s − 0.213·22-s − 1.04·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{5610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−17.88955845178535, −17.34661235428404, −16.52675054043933, −16.22494744175284, −15.54768981492366, −14.81267708022179, −14.39105508100177, −13.49503886510871, −13.14940861919748, −12.43606342792877, −11.60529831138560, −10.99436709591826, −10.20094316088454, −9.897102224753626, −8.943812868969812, −8.631038735267275, −8.023286589876102, −7.076019212259674, −6.458737144869363, −5.913909546232127, −4.909187867285417, −3.652116232586973, −3.396138240836892, −1.993166114878818, −1.590196852226488, 0, 1.590196852226488, 1.993166114878818, 3.396138240836892, 3.652116232586973, 4.909187867285417, 5.913909546232127, 6.458737144869363, 7.076019212259674, 8.023286589876102, 8.631038735267275, 8.943812868969812, 9.897102224753626, 10.20094316088454, 10.99436709591826, 11.60529831138560, 12.43606342792877, 13.14940861919748, 13.49503886510871, 14.39105508100177, 14.81267708022179, 15.54768981492366, 16.22494744175284, 16.52675054043933, 17.34661235428404, 17.88955845178535

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