Properties

Degree $2$
Conductor $5610$
Sign $1$
Motivic weight $1$
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 − 2·7-s − 8-s + 9-s + 10-s − 11-s − 12-s + 2·14-s + 15-s + 16-s + 17-s − 18-s − 8·19-s − 20-s + 2·21-s + 22-s − 4·23-s + 24-s + 25-s − 27-s − 2·28-s − 6·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.182·30-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: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3279780689\)
\(L(\frac12)\) \(\approx\) \(0.3279780689\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−17.51619049348146, −16.93588578739276, −16.38563857030534, −15.98604391694566, −15.28350417176991, −14.81028582747798, −13.95237817067286, −13.00733823732655, −12.61439537735672, −12.09179530798950, −11.21979517949884, −10.76175499519389, −10.23671976274526, −9.464999227772764, −8.889685558149378, −8.091602130399179, −7.458728670763948, −6.778584826898829, −6.080553738434149, −5.524087103624119, −4.343884375727737, −3.749363046821923, −2.662300148363520, −1.735757987764200, −0.3379229914357529, 0.3379229914357529, 1.735757987764200, 2.662300148363520, 3.749363046821923, 4.343884375727737, 5.524087103624119, 6.080553738434149, 6.778584826898829, 7.458728670763948, 8.091602130399179, 8.889685558149378, 9.464999227772764, 10.23671976274526, 10.76175499519389, 11.21979517949884, 12.09179530798950, 12.61439537735672, 13.00733823732655, 13.95237817067286, 14.81028582747798, 15.28350417176991, 15.98604391694566, 16.38563857030534, 16.93588578739276, 17.51619049348146

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