Properties

Label 2-450-1.1-c5-0-36
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 − 17.4·7-s + 64·8-s + 645.·11-s − 1.09e3·13-s − 69.7·14-s + 256·16-s − 1.16e3·17-s − 2.24e3·19-s + 2.58e3·22-s − 500·23-s − 4.39e3·26-s − 278.·28-s − 470.·29-s + 3.85e3·31-s + 1.02e3·32-s − 4.66e3·34-s + 6.99e3·37-s − 8.97e3·38-s + 9.58e3·41-s − 5.30e3·43-s + 1.03e4·44-s − 2.00e3·46-s − 1.99e4·47-s − 1.65e4·49-s − 1.75e4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.134·7-s + 0.353·8-s + 1.60·11-s − 1.80·13-s − 0.0950·14-s + 0.250·16-s − 0.978·17-s − 1.42·19-s + 1.13·22-s − 0.197·23-s − 1.27·26-s − 0.0672·28-s − 0.103·29-s + 0.720·31-s + 0.176·32-s − 0.691·34-s + 0.839·37-s − 1.00·38-s + 0.890·41-s − 0.437·43-s + 0.803·44-s − 0.139·46-s − 1.31·47-s − 0.981·49-s − 0.901·52-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: Trivial
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 \( 1 + 17.4T + 1.68e4T^{2} \)
11 \( 1 - 645.T + 1.61e5T^{2} \)
13 \( 1 + 1.09e3T + 3.71e5T^{2} \)
17 \( 1 + 1.16e3T + 1.41e6T^{2} \)
19 \( 1 + 2.24e3T + 2.47e6T^{2} \)
23 \( 1 + 500T + 6.43e6T^{2} \)
29 \( 1 + 470.T + 2.05e7T^{2} \)
31 \( 1 - 3.85e3T + 2.86e7T^{2} \)
37 \( 1 - 6.99e3T + 6.93e7T^{2} \)
41 \( 1 - 9.58e3T + 1.15e8T^{2} \)
43 \( 1 + 5.30e3T + 1.47e8T^{2} \)
47 \( 1 + 1.99e4T + 2.29e8T^{2} \)
53 \( 1 + 1.14e3T + 4.18e8T^{2} \)
59 \( 1 + 3.73e4T + 7.14e8T^{2} \)
61 \( 1 + 3.81e4T + 8.44e8T^{2} \)
67 \( 1 + 3.62e4T + 1.35e9T^{2} \)
71 \( 1 + 1.66e4T + 1.80e9T^{2} \)
73 \( 1 - 6.93e4T + 2.07e9T^{2} \)
79 \( 1 + 2.06e4T + 3.07e9T^{2} \)
83 \( 1 + 9.69e4T + 3.93e9T^{2} \)
89 \( 1 - 5.96e4T + 5.58e9T^{2} \)
97 \( 1 - 5.82e3T + 8.58e9T^{2} \)
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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

−9.818614998055517722764349907346, −9.061930879948732781601743855471, −7.85234456004308242208786877314, −6.73737122095056696438801894178, −6.23568701966129655737109488399, −4.74132969461172233351228165437, −4.18241215342490192383373159227, −2.79060703526177664011971312297, −1.71923132028723569310754104046, 0, 1.71923132028723569310754104046, 2.79060703526177664011971312297, 4.18241215342490192383373159227, 4.74132969461172233351228165437, 6.23568701966129655737109488399, 6.73737122095056696438801894178, 7.85234456004308242208786877314, 9.061930879948732781601743855471, 9.818614998055517722764349907346

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