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 − 7·13-s + 16-s − 4·17-s − 19-s + 20-s − 22-s − 23-s + 25-s + 7·26-s + 8·29-s − 6·31-s − 32-s + 4·34-s − 3·37-s + 38-s − 40-s + 9·41-s − 4·43-s + 44-s + 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 + 0.301·11-s − 1.94·13-s + 1/4·16-s − 0.970·17-s − 0.229·19-s + 0.223·20-s − 0.213·22-s − 0.208·23-s + 1/5·25-s + 1.37·26-s + 1.48·29-s − 1.07·31-s − 0.176·32-s + 0.685·34-s − 0.493·37-s + 0.162·38-s − 0.158·40-s + 1.40·41-s − 0.609·43-s + 0.150·44-s + 0.147·46-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: $\chi_{4410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.124210926\)
\(L(\frac12)\) \(\approx\) \(1.124210926\)
\(L(\frac{3}{2})\) 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 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−8.396627832349766845018692016394, −7.69234677254200114708354751195, −6.84239878685049028219148372324, −6.52058887037352170698991748319, −5.36057786099946828982465209074, −4.78420387569213441370931648811, −3.72650469584049527071254512684, −2.48128176333592537039367528987, −2.09100988764213577802437323995, −0.63547758295212514566981031432, 0.63547758295212514566981031432, 2.09100988764213577802437323995, 2.48128176333592537039367528987, 3.72650469584049527071254512684, 4.78420387569213441370931648811, 5.36057786099946828982465209074, 6.52058887037352170698991748319, 6.84239878685049028219148372324, 7.69234677254200114708354751195, 8.396627832349766845018692016394

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