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 − 8-s + 9-s − 10-s − 11-s − 12-s + 2·13-s − 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s + 22-s + 8·23-s + 24-s + 25-s − 2·26-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 + 0.554·13-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 + 0.213·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-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: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.331886409\)
\(L(\frac12)\) \(\approx\) \(1.331886409\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−17.59196698002720, −17.02906599700336, −16.50774164624077, −15.76214965209167, −15.48851758683854, −14.62051665601689, −13.85165966050604, −13.27858401753662, −12.64595471204521, −11.88323446837922, −11.33528729532178, −10.72479106633623, −10.14781231851425, −9.564415400159329, −8.806346478273130, −8.263245916652138, −7.352842891348102, −6.730340229836768, −6.179797301058110, −5.247820210113309, −4.784456161983381, −3.477223248658312, −2.725584527512967, −1.571475258577874, −0.7504924848051875, 0.7504924848051875, 1.571475258577874, 2.725584527512967, 3.477223248658312, 4.784456161983381, 5.247820210113309, 6.179797301058110, 6.730340229836768, 7.352842891348102, 8.263245916652138, 8.806346478273130, 9.564415400159329, 10.14781231851425, 10.72479106633623, 11.33528729532178, 11.88323446837922, 12.64595471204521, 13.27858401753662, 13.85165966050604, 14.62051665601689, 15.48851758683854, 15.76214965209167, 16.50774164624077, 17.02906599700336, 17.59196698002720

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