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 + 6·11-s + 4·13-s + 16-s + 6·17-s + 4·19-s − 20-s − 6·22-s + 25-s − 4·26-s − 6·29-s + 4·31-s − 32-s − 6·34-s + 8·37-s − 4·38-s + 40-s + 8·43-s + 6·44-s − 50-s + 4·52-s − 6·53-s − 6·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.80·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.784·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.31·37-s − 0.648·38-s + 0.158·40-s + 1.21·43-s + 0.904·44-s − 0.141·50-s + 0.554·52-s − 0.824·53-s − 0.809·55-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

\(L(1)\) \(\approx\) \(1.733127622\)
\(L(\frac12)\) \(\approx\) \(1.733127622\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−18.03727962773670, −17.25716562425285, −16.70911134324228, −16.33630216337884, −15.61981814776391, −14.96246076507261, −14.27040466227733, −13.87974167138147, −12.81204067898350, −12.20707410051373, −11.45108358839021, −11.31185266555420, −10.31170758173722, −9.521684969964858, −9.171828348816100, −8.361683788765927, −7.680725666871873, −7.100979960373113, −6.150955606015013, −5.763067992268800, −4.429008897021144, −3.679176550747375, −3.020775548552348, −1.502580677258279, −0.9459582071878224, 0.9459582071878224, 1.502580677258279, 3.020775548552348, 3.679176550747375, 4.429008897021144, 5.763067992268800, 6.150955606015013, 7.100979960373113, 7.680725666871873, 8.361683788765927, 9.171828348816100, 9.521684969964858, 10.31170758173722, 11.31185266555420, 11.45108358839021, 12.20707410051373, 12.81204067898350, 13.87974167138147, 14.27040466227733, 14.96246076507261, 15.61981814776391, 16.33630216337884, 16.70911134324228, 17.25716562425285, 18.03727962773670

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