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 + 13-s + 3·14-s − 15-s + 16-s − 17-s − 18-s − 3·19-s − 20-s − 3·21-s − 22-s − 23-s − 24-s + 25-s − 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 + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.688·19-s − 0.223·20-s − 0.654·21-s − 0.213·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-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 \( 1 - T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−17.99698315544976, −17.22884444819909, −16.52119957481909, −16.09966708350847, −15.59757270521175, −14.97194619880446, −14.35035952785537, −13.60343205011511, −12.87003748505999, −12.50676856894910, −11.64022140355574, −11.06570865662625, −10.19660370112403, −9.827358139148567, −9.055500364925005, −8.562688314256250, −7.913531864345150, −7.156513212424404, −6.475353497677693, −6.014717988329129, −4.677849082196839, −3.909874756511681, −3.112029256409959, −2.435860977403552, −1.220565165229966, 0, 1.220565165229966, 2.435860977403552, 3.112029256409959, 3.909874756511681, 4.677849082196839, 6.014717988329129, 6.475353497677693, 7.156513212424404, 7.913531864345150, 8.562688314256250, 9.055500364925005, 9.827358139148567, 10.19660370112403, 11.06570865662625, 11.64022140355574, 12.50676856894910, 12.87003748505999, 13.60343205011511, 14.35035952785537, 14.97194619880446, 15.59757270521175, 16.09966708350847, 16.52119957481909, 17.22884444819909, 17.99698315544976

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