Properties

Label 2-450-1.1-c5-0-3
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s − 141.·7-s − 64·8-s − 113.·11-s − 61.7·13-s + 566.·14-s + 256·16-s − 1.67e3·17-s − 662.·19-s + 453.·22-s − 86.4·23-s + 246.·26-s − 2.26e3·28-s + 3.23e3·29-s − 3.81e3·31-s − 1.02e3·32-s + 6.68e3·34-s − 1.02e4·37-s + 2.64e3·38-s + 1.34e4·41-s + 4.69e3·43-s − 1.81e3·44-s + 345.·46-s + 1.52e4·47-s + 3.25e3·49-s − 987.·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.09·7-s − 0.353·8-s − 0.282·11-s − 0.101·13-s + 0.772·14-s + 0.250·16-s − 1.40·17-s − 0.420·19-s + 0.199·22-s − 0.0340·23-s + 0.0716·26-s − 0.546·28-s + 0.713·29-s − 0.712·31-s − 0.176·32-s + 0.991·34-s − 1.23·37-s + 0.297·38-s + 1.25·41-s + 0.387·43-s − 0.141·44-s + 0.0241·46-s + 1.00·47-s + 0.193·49-s − 0.0506·52-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: \(0\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7052849715\)
\(L(\frac12)\) \(\approx\) \(0.7052849715\)
\(L(\frac{7}{2})\) 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 \( 1 + 141.T + 1.68e4T^{2} \)
11 \( 1 + 113.T + 1.61e5T^{2} \)
13 \( 1 + 61.7T + 3.71e5T^{2} \)
17 \( 1 + 1.67e3T + 1.41e6T^{2} \)
19 \( 1 + 662.T + 2.47e6T^{2} \)
23 \( 1 + 86.4T + 6.43e6T^{2} \)
29 \( 1 - 3.23e3T + 2.05e7T^{2} \)
31 \( 1 + 3.81e3T + 2.86e7T^{2} \)
37 \( 1 + 1.02e4T + 6.93e7T^{2} \)
41 \( 1 - 1.34e4T + 1.15e8T^{2} \)
43 \( 1 - 4.69e3T + 1.47e8T^{2} \)
47 \( 1 - 1.52e4T + 2.29e8T^{2} \)
53 \( 1 + 498.T + 4.18e8T^{2} \)
59 \( 1 - 1.52e4T + 7.14e8T^{2} \)
61 \( 1 + 3.18e4T + 8.44e8T^{2} \)
67 \( 1 + 4.92e4T + 1.35e9T^{2} \)
71 \( 1 + 2.10e4T + 1.80e9T^{2} \)
73 \( 1 + 3.94e4T + 2.07e9T^{2} \)
79 \( 1 - 7.29e4T + 3.07e9T^{2} \)
83 \( 1 + 1.00e5T + 3.93e9T^{2} \)
89 \( 1 - 1.46e5T + 5.58e9T^{2} \)
97 \( 1 - 4.35e4T + 8.58e9T^{2} \)
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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

−10.30430860287856536412095920278, −9.257374934401568535077012748577, −8.734813616639824872696948552561, −7.50685482881525271668780725221, −6.68709668876032035040045315502, −5.86098021955007628672792460913, −4.40876468403547833817143894168, −3.11754184772086575357561348217, −2.06248025305415981788920729837, −0.45735918814185854214405374326, 0.45735918814185854214405374326, 2.06248025305415981788920729837, 3.11754184772086575357561348217, 4.40876468403547833817143894168, 5.86098021955007628672792460913, 6.68709668876032035040045315502, 7.50685482881525271668780725221, 8.734813616639824872696948552561, 9.257374934401568535077012748577, 10.30430860287856536412095920278

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