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 − 3·7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 4·13-s + 3·14-s − 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s + 3·21-s + 22-s − 23-s + 24-s + 25-s + 4·26-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 − 1.10·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-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 + 0.784·26-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: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5818909762\)
\(L(\frac12)\) \(\approx\) \(0.5818909762\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−17.57573303431082, −16.95412011196166, −16.46815959422131, −15.88520952753173, −15.48711504201561, −14.53826826620936, −13.99433050410750, −13.08931469348708, −12.57413459419791, −12.18112465628396, −11.20229450237686, −10.78083266858666, −9.949469239252397, −9.517151171834578, −9.185030592918746, −8.023692370246824, −7.337078469973334, −6.841194797516550, −6.021570024204847, −5.498513183995996, −4.624628778293063, −3.479193989464501, −2.695492678683961, −1.757983523359926, −0.4506786160172048, 0.4506786160172048, 1.757983523359926, 2.695492678683961, 3.479193989464501, 4.624628778293063, 5.498513183995996, 6.021570024204847, 6.841194797516550, 7.337078469973334, 8.023692370246824, 9.185030592918746, 9.517151171834578, 9.949469239252397, 10.78083266858666, 11.20229450237686, 12.18112465628396, 12.57413459419791, 13.08931469348708, 13.99433050410750, 14.53826826620936, 15.48711504201561, 15.88520952753173, 16.46815959422131, 16.95412011196166, 17.57573303431082

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