Properties

Degree $2$
Conductor $4410$
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 + 4-s + 5-s − 8-s − 10-s − 3·11-s + 5·13-s + 16-s − 6·17-s − 19-s + 20-s + 3·22-s − 3·23-s + 25-s − 5·26-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 11·37-s + 38-s − 40-s − 3·41-s − 10·43-s − 3·44-s + 3·46-s − 3·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.904·11-s + 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.229·19-s + 0.223·20-s + 0.639·22-s − 0.625·23-s + 1/5·25-s − 0.980·26-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.468·41-s − 1.52·43-s − 0.452·44-s + 0.442·46-s − 0.437·47-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: \(1\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−18.16395441027803, −17.91361660150350, −17.11727182272056, −16.37083490706844, −15.93738164227452, −15.39653496879724, −14.70946386937248, −13.85229174106804, −13.14587523324103, −12.98450448413232, −11.77769008294831, −11.29645844641002, −10.55802549411330, −10.19839182231822, −9.289477817272518, −8.675058805437000, −8.179612029398430, −7.390347999883002, −6.352588588093976, −6.183311392662976, −5.075185265340129, −4.228105526304225, −3.139927008139655, −2.284871326009367, −1.384453095916702, 0, 1.384453095916702, 2.284871326009367, 3.139927008139655, 4.228105526304225, 5.075185265340129, 6.183311392662976, 6.352588588093976, 7.390347999883002, 8.179612029398430, 8.675058805437000, 9.289477817272518, 10.19839182231822, 10.55802549411330, 11.29645844641002, 11.77769008294831, 12.98450448413232, 13.14587523324103, 13.85229174106804, 14.70946386937248, 15.39653496879724, 15.93738164227452, 16.37083490706844, 17.11727182272056, 17.91361660150350, 18.16395441027803

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