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 − 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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
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

−8.100703075007619597471465975056, −7.34551420560314008593164394824, −6.63808060505647498349846702623, −5.91639559154269628299836756808, −5.11071797962091529492876313221, −4.28094102748731103378935315617, −3.04050596042898199200679637351, −2.37705903944982513520184893200, −1.34876055133461141882873475535, 0, 1.34876055133461141882873475535, 2.37705903944982513520184893200, 3.04050596042898199200679637351, 4.28094102748731103378935315617, 5.11071797962091529492876313221, 5.91639559154269628299836756808, 6.63808060505647498349846702623, 7.34551420560314008593164394824, 8.100703075007619597471465975056

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