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 + 2·11-s + 16-s − 4·17-s + 6·19-s − 20-s − 2·22-s − 3·23-s + 25-s − 9·29-s + 4·31-s − 32-s + 4·34-s − 4·37-s − 6·38-s + 40-s − 7·41-s − 5·43-s + 2·44-s + 3·46-s + 8·47-s − 50-s + 2·53-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 + 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.973·38-s + 0.158·40-s − 1.09·41-s − 0.762·43-s + 0.301·44-s + 0.442·46-s + 1.16·47-s − 0.141·50-s + 0.274·53-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.065939793239333672640685356709, −7.26894921723600446928508659816, −6.81744244857954183150027466301, −5.89588864620231652033142375263, −5.09565695139434475827330528759, −4.05084036592834491429461078310, −3.37174455571566278352801880800, −2.27179767436923261006533101482, −1.28265228999086935733577332206, 0, 1.28265228999086935733577332206, 2.27179767436923261006533101482, 3.37174455571566278352801880800, 4.05084036592834491429461078310, 5.09565695139434475827330528759, 5.89588864620231652033142375263, 6.81744244857954183150027466301, 7.26894921723600446928508659816, 8.065939793239333672640685356709

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