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 − 3·11-s + 13-s + 16-s − 6·17-s + 19-s + 20-s − 3·22-s − 9·23-s + 25-s + 26-s − 6·29-s − 8·31-s + 32-s − 6·34-s − 7·37-s + 38-s + 40-s + 3·41-s + 2·43-s − 3·44-s − 9·46-s + 9·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.229·19-s + 0.223·20-s − 0.639·22-s − 1.87·23-s + 1/5·25-s + 0.196·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 1.15·37-s + 0.162·38-s + 0.158·40-s + 0.468·41-s + 0.304·43-s − 0.452·44-s − 1.32·46-s + 1.31·47-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: $\chi_{4410} (1, \cdot )$
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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.80429629873338082381865447532, −7.25804679311028595514768686057, −6.31873949780136891248251287978, −5.76691681189958170990642289846, −5.08363253751508031682005126897, −4.20473169440760120227322418457, −3.48933454534763160692000672822, −2.35777536635960010918119064141, −1.82372761545219197495604647685, 0, 1.82372761545219197495604647685, 2.35777536635960010918119064141, 3.48933454534763160692000672822, 4.20473169440760120227322418457, 5.08363253751508031682005126897, 5.76691681189958170990642289846, 6.31873949780136891248251287978, 7.25804679311028595514768686057, 7.80429629873338082381865447532

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