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 − 2·11-s − 2·13-s + 16-s + 2·17-s − 6·19-s + 20-s + 2·22-s + 4·23-s + 25-s + 2·26-s + 2·31-s − 32-s − 2·34-s + 2·37-s + 6·38-s − 40-s + 10·41-s − 8·43-s − 2·44-s − 4·46-s − 8·47-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 − 0.603·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.223·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 0.359·31-s − 0.176·32-s − 0.342·34-s + 0.328·37-s + 0.973·38-s − 0.158·40-s + 1.56·41-s − 1.21·43-s − 0.301·44-s − 0.589·46-s − 1.16·47-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−18.09138045445869, −17.76078983437634, −16.98144933409216, −16.64541651711175, −15.97955015752272, −15.03470459284323, −14.88123327152499, −14.00664569253367, −13.16155443041039, −12.73292157423847, −12.01637615560362, −11.17315415916786, −10.65235936676903, −9.996151731852235, −9.474059143661811, −8.667947818270754, −8.100703075007620, −7.345514205603140, −6.638080605056475, −5.916395591542696, −5.110717979620915, −4.280941027487311, −3.040505960428982, −2.377059039449825, −1.348760551334611, 0, 1.348760551334611, 2.377059039449825, 3.040505960428982, 4.280941027487311, 5.110717979620915, 5.916395591542696, 6.638080605056475, 7.345514205603140, 8.100703075007620, 8.667947818270754, 9.474059143661811, 9.996151731852235, 10.65235936676903, 11.17315415916786, 12.01637615560362, 12.73292157423847, 13.16155443041039, 14.00664569253367, 14.88123327152499, 15.03470459284323, 15.97955015752272, 16.64541651711175, 16.98144933409216, 17.76078983437634, 18.09138045445869

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