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 − 4·11-s − 6·13-s + 16-s − 4·17-s − 6·19-s − 20-s + 4·22-s + 25-s + 6·26-s − 6·29-s + 4·31-s − 32-s + 4·34-s + 8·37-s + 6·38-s + 40-s − 10·41-s − 2·43-s − 4·44-s − 10·47-s − 50-s − 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 1/4·16-s − 0.970·17-s − 1.37·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s + 1.31·37-s + 0.973·38-s + 0.158·40-s − 1.56·41-s − 0.304·43-s − 0.603·44-s − 1.45·47-s − 0.141·50-s − 0.832·52-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: $\chi_{4410} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4266182995\)
\(L(\frac12)\) \(\approx\) \(0.4266182995\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−18.07339379229175, −17.25810887960074, −16.83946667337719, −16.26447418719222, −15.41670866690843, −15.03462586298108, −14.63219794709300, −13.38435081687400, −13.02636842737127, −12.28810853145370, −11.59981614224381, −11.00171066929570, −10.27937580448064, −9.837511959604680, −9.004622421249755, −8.218003904598524, −7.832612366939416, −6.986530874320883, −6.476358752320349, −5.295707057582443, −4.743869203290660, −3.750631512444107, −2.524078459571864, −2.149201776331577, −0.3762582939476479, 0.3762582939476479, 2.149201776331577, 2.524078459571864, 3.750631512444107, 4.743869203290660, 5.295707057582443, 6.476358752320349, 6.986530874320883, 7.832612366939416, 8.218003904598524, 9.004622421249755, 9.837511959604680, 10.27937580448064, 11.00171066929570, 11.59981614224381, 12.28810853145370, 13.02636842737127, 13.38435081687400, 14.63219794709300, 15.03462586298108, 15.41670866690843, 16.26447418719222, 16.83946667337719, 17.25810887960074, 18.07339379229175

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