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 + 11-s + 13-s + 16-s − 3·19-s + 20-s + 22-s − 7·23-s + 25-s + 26-s + 8·29-s − 2·31-s + 32-s + 11·37-s − 3·38-s + 40-s + 11·41-s + 8·43-s + 44-s − 7·46-s + 5·47-s + 50-s + 52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.688·19-s + 0.223·20-s + 0.213·22-s − 1.45·23-s + 1/5·25-s + 0.196·26-s + 1.48·29-s − 0.359·31-s + 0.176·32-s + 1.80·37-s − 0.486·38-s + 0.158·40-s + 1.71·41-s + 1.21·43-s + 0.150·44-s − 1.03·46-s + 0.729·47-s + 0.141·50-s + 0.138·52-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: $\chi_{4410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.556997826\)
\(L(\frac12)\) \(\approx\) \(3.556997826\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.264482922146465648328712134249, −7.57805661572748328996357007125, −6.69013777704350789870596700067, −6.01576612965313135710541185703, −5.60913615063324456328754249658, −4.31077665082661801387980471249, −4.15209906358649170610383768387, −2.83104229741502313280721004942, −2.19103638269681867060894200237, −0.984476044023856808493085301651, 0.984476044023856808493085301651, 2.19103638269681867060894200237, 2.83104229741502313280721004942, 4.15209906358649170610383768387, 4.31077665082661801387980471249, 5.60913615063324456328754249658, 6.01576612965313135710541185703, 6.69013777704350789870596700067, 7.57805661572748328996357007125, 8.264482922146465648328712134249

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