Properties

Label 2-4410-1.1-c1-0-54
Degree $2$
Conductor $4410$
Sign $-1$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
Motivic weight $1$
Arithmetic yes
Rational yes
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: $-1$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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

−7.978867865315999140103486121694, −7.43085085363014212731281452820, −6.51215653279629143098878255567, −5.99869648860145192358302637159, −5.18403001049890241512229669344, −4.08226266820595925481841090344, −3.30779312858222897350179871072, −2.13341818101494808635243606638, −1.46450058564780580279055069521, 0, 1.46450058564780580279055069521, 2.13341818101494808635243606638, 3.30779312858222897350179871072, 4.08226266820595925481841090344, 5.18403001049890241512229669344, 5.99869648860145192358302637159, 6.51215653279629143098878255567, 7.43085085363014212731281452820, 7.978867865315999140103486121694

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