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

\(L(1)\) \(\approx\) \(2.868370817\)
\(L(\frac12)\) \(\approx\) \(2.868370817\)
\(L(\frac{3}{2})\) 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

−8.150149311642982143720753419665, −7.64520969807885352996375891508, −6.76465999798410197970364155905, −6.17955291083753433044886873040, −5.33029838161233124836993267190, −4.53259138208873278855056003349, −3.90593184920529201683720597533, −3.03073886106815598508910759478, −2.15666973839873817490152762248, −0.864051942346481283386568118080, 0.864051942346481283386568118080, 2.15666973839873817490152762248, 3.03073886106815598508910759478, 3.90593184920529201683720597533, 4.53259138208873278855056003349, 5.33029838161233124836993267190, 6.17955291083753433044886873040, 6.76465999798410197970364155905, 7.64520969807885352996375891508, 8.150149311642982143720753419665

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