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 − 3·7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 5·13-s + 3·14-s − 15-s + 16-s + 17-s − 18-s + 7·19-s − 20-s − 3·21-s + 22-s + 7·23-s − 24-s + 25-s + 5·26-s + 27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-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 + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.60·19-s − 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.980·26-s + 0.192·27-s − 0.566·28-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−18.03679818680079, −17.09626708857060, −16.61233227579964, −16.18024597661220, −15.47545125931791, −14.91568217412483, −14.52458380979549, −13.32159897067419, −13.20179317391636, −12.26584415275779, −11.79069674721101, −11.06349660791788, −10.14676303749824, −9.643283670861856, −9.389440132501351, −8.455133426689808, −7.792878192315735, −7.091352815637659, −6.856846935225423, −5.611447259732191, −4.959664905777766, −3.799704862528877, −3.014220335810514, −2.573564754677018, −1.173436902601911, 0, 1.173436902601911, 2.573564754677018, 3.014220335810514, 3.799704862528877, 4.959664905777766, 5.611447259732191, 6.856846935225423, 7.091352815637659, 7.792878192315735, 8.455133426689808, 9.389440132501351, 9.643283670861856, 10.14676303749824, 11.06349660791788, 11.79069674721101, 12.26584415275779, 13.20179317391636, 13.32159897067419, 14.52458380979549, 14.91568217412483, 15.47545125931791, 16.18024597661220, 16.61233227579964, 17.09626708857060, 18.03679818680079

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