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 − 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: $\chi_{4410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.109031088\)
\(L(\frac12)\) \(\approx\) \(1.109031088\)
\(L(\frac{3}{2})\) 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

−8.534223993894932133935300439508, −7.54989395954188160037997215118, −7.20528475175092734810498968650, −6.30835453368910666484606355248, −5.43872560400832123096403739032, −4.71215531335858462495325223484, −3.55974650518840309017563415718, −2.93117589735157766666354392580, −1.77148799114426370056808540256, −0.66281130095645621330697335573, 0.66281130095645621330697335573, 1.77148799114426370056808540256, 2.93117589735157766666354392580, 3.55974650518840309017563415718, 4.71215531335858462495325223484, 5.43872560400832123096403739032, 6.30835453368910666484606355248, 7.20528475175092734810498968650, 7.54989395954188160037997215118, 8.534223993894932133935300439508

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