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 + 4·11-s + 2·13-s + 16-s − 8·17-s − 6·19-s − 20-s − 4·22-s + 4·23-s + 25-s − 2·26-s + 6·29-s + 4·31-s − 32-s + 8·34-s − 10·37-s + 6·38-s + 40-s − 4·41-s + 4·43-s + 4·44-s − 4·46-s − 4·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 1.94·17-s − 1.37·19-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.37·34-s − 1.64·37-s + 0.973·38-s + 0.158·40-s − 0.624·41-s + 0.609·43-s + 0.603·44-s − 0.589·46-s − 0.583·47-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−18.12230904945023, −17.58238657301143, −17.17396140876616, −16.55719837011255, −15.65672902607256, −15.49418071866840, −14.72256939915346, −14.01370470824550, −13.27545441769713, −12.59638949233282, −11.86311134603910, −11.29314267973794, −10.75966542850478, −10.13570528209283, −9.065792963445518, −8.764915580044923, −8.291052151464846, −7.178684019965694, −6.558888697466805, −6.269500244783961, −4.832135164014472, −4.227723763085172, −3.324831966945759, −2.255053608530153, −1.299308092107439, 0, 1.299308092107439, 2.255053608530153, 3.324831966945759, 4.227723763085172, 4.832135164014472, 6.269500244783961, 6.558888697466805, 7.178684019965694, 8.291052151464846, 8.764915580044923, 9.065792963445518, 10.13570528209283, 10.75966542850478, 11.29314267973794, 11.86311134603910, 12.59638949233282, 13.27545441769713, 14.01370470824550, 14.72256939915346, 15.49418071866840, 15.65672902607256, 16.55719837011255, 17.17396140876616, 17.58238657301143, 18.12230904945023

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