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 + 5·11-s − 5·13-s + 16-s + 4·17-s − 7·19-s − 20-s − 5·22-s − 23-s + 25-s + 5·26-s − 2·31-s − 32-s − 4·34-s + 37-s + 7·38-s + 40-s − 5·41-s + 12·43-s + 5·44-s + 46-s + 11·47-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.50·11-s − 1.38·13-s + 1/4·16-s + 0.970·17-s − 1.60·19-s − 0.223·20-s − 1.06·22-s − 0.208·23-s + 1/5·25-s + 0.980·26-s − 0.359·31-s − 0.176·32-s − 0.685·34-s + 0.164·37-s + 1.13·38-s + 0.158·40-s − 0.780·41-s + 1.82·43-s + 0.753·44-s + 0.147·46-s + 1.60·47-s − 0.141·50-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

\(L(1)\) \(\approx\) \(1.107026500\)
\(L(\frac12)\) \(\approx\) \(1.107026500\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−17.82005773078061, −17.21151481612525, −16.83269112552816, −16.44096394713364, −15.43771165929755, −14.91226974753062, −14.48492399241863, −13.82960365080436, −12.62953136410418, −12.27810897602666, −11.77971433007586, −10.98260982986972, −10.33268486025375, −9.676167092885010, −9.003917863434136, −8.484879927600426, −7.537871617844934, −7.146497501478420, −6.325408750641715, −5.577753176610165, −4.422804027739166, −3.870802759797380, −2.744869863798505, −1.831351979607969, −0.6559171145780803, 0.6559171145780803, 1.831351979607969, 2.744869863798505, 3.870802759797380, 4.422804027739166, 5.577753176610165, 6.325408750641715, 7.146497501478420, 7.537871617844934, 8.484879927600426, 9.003917863434136, 9.676167092885010, 10.33268486025375, 10.98260982986972, 11.77971433007586, 12.27810897602666, 12.62953136410418, 13.82960365080436, 14.48492399241863, 14.91226974753062, 15.43771165929755, 16.44096394713364, 16.83269112552816, 17.21151481612525, 17.82005773078061

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