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 + 2·7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 17-s − 18-s + 2·19-s − 20-s + 2·21-s − 22-s − 6·23-s − 24-s + 25-s + 4·26-s + 27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.377·28-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.84057031617058, −17.40660044205534, −16.66897984972678, −16.02026944637498, −15.54933934230584, −14.77633707155404, −14.42070571762278, −13.82361525426473, −12.95180768066819, −12.03737660377247, −11.88890385470289, −11.09141177196516, −10.14302528905099, −9.985006428065826, −8.903242829359164, −8.569438660830790, −7.814950005496811, −7.314313469408219, −6.697774273244272, −5.622810860180895, −4.793244751185757, −4.038415003061634, −3.100884855399898, −2.210226272387147, −1.410400715078571, 0, 1.410400715078571, 2.210226272387147, 3.100884855399898, 4.038415003061634, 4.793244751185757, 5.622810860180895, 6.697774273244272, 7.314313469408219, 7.814950005496811, 8.569438660830790, 8.903242829359164, 9.985006428065826, 10.14302528905099, 11.09141177196516, 11.88890385470289, 12.03737660377247, 12.95180768066819, 13.82361525426473, 14.42070571762278, 14.77633707155404, 15.54933934230584, 16.02026944637498, 16.66897984972678, 17.40660044205534, 17.84057031617058

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