Properties

Degree $2$
Conductor $4410$
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 + 4-s + 5-s − 8-s − 10-s − 2·11-s − 2·13-s + 16-s − 4·17-s + 20-s + 2·22-s − 8·23-s + 25-s + 2·26-s + 2·31-s − 32-s + 4·34-s + 8·37-s − 40-s − 2·41-s − 2·43-s − 2·44-s + 8·46-s + 10·47-s − 50-s − 2·52-s + 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s + 1/4·16-s − 0.970·17-s + 0.223·20-s + 0.426·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.359·31-s − 0.176·32-s + 0.685·34-s + 1.31·37-s − 0.158·40-s − 0.312·41-s − 0.304·43-s − 0.301·44-s + 1.17·46-s + 1.45·47-s − 0.141·50-s − 0.277·52-s + 0.274·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4410} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.113377189\)
\(L(\frac12)\) \(\approx\) \(1.113377189\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−17.86981972755233, −17.55027023488348, −16.76427103397192, −16.21723738615113, −15.61595243534585, −15.01946133664481, −14.30988212396293, −13.58987054961803, −13.05623859223763, −12.22194727759293, −11.70343047765563, −10.86118794713070, −10.33286019437901, −9.705638590350501, −9.153631741535801, −8.275900421424892, −7.818010139884511, −6.951599381699962, −6.279852747921275, −5.554925050296559, −4.675961565710652, −3.747547596784865, −2.484414116231922, −2.088688288558272, −0.6321108860456753, 0.6321108860456753, 2.088688288558272, 2.484414116231922, 3.747547596784865, 4.675961565710652, 5.554925050296559, 6.279852747921275, 6.951599381699962, 7.818010139884511, 8.275900421424892, 9.153631741535801, 9.705638590350501, 10.33286019437901, 10.86118794713070, 11.70343047765563, 12.22194727759293, 13.05623859223763, 13.58987054961803, 14.30988212396293, 15.01946133664481, 15.61595243534585, 16.21723738615113, 16.76427103397192, 17.55027023488348, 17.86981972755233

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