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 − 8-s + 9-s + 10-s − 11-s − 12-s − 6·13-s + 15-s + 16-s − 17-s − 18-s + 8·19-s − 20-s + 22-s − 8·23-s + 24-s + 25-s + 6·26-s − 27-s + 6·29-s − 30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-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 + 0.213·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−17.84552015312071, −17.37621237725067, −16.66111888031679, −15.99528390263694, −15.83054524655627, −14.98567124187006, −14.29034747113194, −13.72996624222989, −12.66779515444982, −12.25672657038444, −11.60035954172386, −11.29868502888220, −10.21757078678346, −9.858263565664145, −9.429417973391120, −8.197038938003062, −7.876883852712657, −7.183286721525483, −6.528132094568792, −5.655098400307371, −4.935640034627962, −4.213401420756628, −3.038769570223984, −2.304425374270183, −1.030905026661884, 0, 1.030905026661884, 2.304425374270183, 3.038769570223984, 4.213401420756628, 4.935640034627962, 5.655098400307371, 6.528132094568792, 7.183286721525483, 7.876883852712657, 8.197038938003062, 9.429417973391120, 9.858263565664145, 10.21757078678346, 11.29868502888220, 11.60035954172386, 12.25672657038444, 12.66779515444982, 13.72996624222989, 14.29034747113194, 14.98567124187006, 15.83054524655627, 15.99528390263694, 16.66111888031679, 17.37621237725067, 17.84552015312071

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