Properties

Label 2-450-1.1-c5-0-39
Degree $2$
Conductor $450$
Sign $-1$
Analytic cond. $72.1727$
Root an. cond. $8.49545$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 241.·7-s + 64·8-s − 653.·11-s − 828.·13-s + 966.·14-s + 256·16-s − 2.16e3·17-s + 1.25e3·19-s − 2.61e3·22-s − 3.74e3·23-s − 3.31e3·26-s + 3.86e3·28-s − 2.46e3·29-s − 1.89e3·31-s + 1.02e3·32-s − 8.64e3·34-s − 1.05e3·37-s + 5.01e3·38-s − 1.96e3·41-s − 1.10e4·43-s − 1.04e4·44-s − 1.49e4·46-s + 2.30e4·47-s + 4.15e4·49-s − 1.32e4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.86·7-s + 0.353·8-s − 1.62·11-s − 1.35·13-s + 1.31·14-s + 0.250·16-s − 1.81·17-s + 0.797·19-s − 1.15·22-s − 1.47·23-s − 0.961·26-s + 0.931·28-s − 0.544·29-s − 0.354·31-s + 0.176·32-s − 1.28·34-s − 0.126·37-s + 0.563·38-s − 0.182·41-s − 0.908·43-s − 0.813·44-s − 1.04·46-s + 1.52·47-s + 2.47·49-s − 0.679·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(72.1727\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 241.T + 1.68e4T^{2} \)
11 \( 1 + 653.T + 1.61e5T^{2} \)
13 \( 1 + 828.T + 3.71e5T^{2} \)
17 \( 1 + 2.16e3T + 1.41e6T^{2} \)
19 \( 1 - 1.25e3T + 2.47e6T^{2} \)
23 \( 1 + 3.74e3T + 6.43e6T^{2} \)
29 \( 1 + 2.46e3T + 2.05e7T^{2} \)
31 \( 1 + 1.89e3T + 2.86e7T^{2} \)
37 \( 1 + 1.05e3T + 6.93e7T^{2} \)
41 \( 1 + 1.96e3T + 1.15e8T^{2} \)
43 \( 1 + 1.10e4T + 1.47e8T^{2} \)
47 \( 1 - 2.30e4T + 2.29e8T^{2} \)
53 \( 1 - 2.73e4T + 4.18e8T^{2} \)
59 \( 1 + 3.52e4T + 7.14e8T^{2} \)
61 \( 1 - 2.68e3T + 8.44e8T^{2} \)
67 \( 1 + 4.81e4T + 1.35e9T^{2} \)
71 \( 1 + 2.72e4T + 1.80e9T^{2} \)
73 \( 1 - 6.54e3T + 2.07e9T^{2} \)
79 \( 1 + 6.50e4T + 3.07e9T^{2} \)
83 \( 1 - 6.25e4T + 3.93e9T^{2} \)
89 \( 1 + 1.79e4T + 5.58e9T^{2} \)
97 \( 1 + 9.59e4T + 8.58e9T^{2} \)
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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

−10.14899779328195236069171196220, −8.734606535378019109346665991985, −7.75698230636487782360478728166, −7.28380768610713417135931686922, −5.67324472543686894398659633686, −4.95871698913845500016925936815, −4.29500523162034436698643985115, −2.54115146327264989585544592697, −1.88751517892199282527874122681, 0, 1.88751517892199282527874122681, 2.54115146327264989585544592697, 4.29500523162034436698643985115, 4.95871698913845500016925936815, 5.67324472543686894398659633686, 7.28380768610713417135931686922, 7.75698230636487782360478728166, 8.734606535378019109346665991985, 10.14899779328195236069171196220

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