Properties

Label 2-450-1.1-c5-0-14
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 + 119.·7-s + 64·8-s − 263.·11-s − 851.·13-s + 478.·14-s + 256·16-s + 1.28e3·17-s + 2.06e3·19-s − 1.05e3·22-s + 55.5·23-s − 3.40e3·26-s + 1.91e3·28-s + 5.98e3·29-s + 4.78e3·31-s + 1.02e3·32-s + 5.14e3·34-s + 1.21e4·37-s + 8.24e3·38-s − 1.85e4·41-s + 2.18e3·43-s − 4.21e3·44-s + 222.·46-s − 5.59e3·47-s − 2.47e3·49-s − 1.36e4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.923·7-s + 0.353·8-s − 0.657·11-s − 1.39·13-s + 0.652·14-s + 0.250·16-s + 1.08·17-s + 1.30·19-s − 0.464·22-s + 0.0218·23-s − 0.987·26-s + 0.461·28-s + 1.32·29-s + 0.893·31-s + 0.176·32-s + 0.763·34-s + 1.45·37-s + 0.925·38-s − 1.71·41-s + 0.180·43-s − 0.328·44-s + 0.0154·46-s − 0.369·47-s − 0.147·49-s − 0.698·52-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

\(L(3)\) \(\approx\) \(3.868425629\)
\(L(\frac12)\) \(\approx\) \(3.868425629\)
\(L(\frac{7}{2})\) 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 \( 1 - 119.T + 1.68e4T^{2} \)
11 \( 1 + 263.T + 1.61e5T^{2} \)
13 \( 1 + 851.T + 3.71e5T^{2} \)
17 \( 1 - 1.28e3T + 1.41e6T^{2} \)
19 \( 1 - 2.06e3T + 2.47e6T^{2} \)
23 \( 1 - 55.5T + 6.43e6T^{2} \)
29 \( 1 - 5.98e3T + 2.05e7T^{2} \)
31 \( 1 - 4.78e3T + 2.86e7T^{2} \)
37 \( 1 - 1.21e4T + 6.93e7T^{2} \)
41 \( 1 + 1.85e4T + 1.15e8T^{2} \)
43 \( 1 - 2.18e3T + 1.47e8T^{2} \)
47 \( 1 + 5.59e3T + 2.29e8T^{2} \)
53 \( 1 + 2.64e4T + 4.18e8T^{2} \)
59 \( 1 + 2.08e4T + 7.14e8T^{2} \)
61 \( 1 - 4.55e4T + 8.44e8T^{2} \)
67 \( 1 - 3.43e4T + 1.35e9T^{2} \)
71 \( 1 - 5.74e4T + 1.80e9T^{2} \)
73 \( 1 - 2.69e4T + 2.07e9T^{2} \)
79 \( 1 - 4.20e4T + 3.07e9T^{2} \)
83 \( 1 - 1.01e5T + 3.93e9T^{2} \)
89 \( 1 - 6.55e4T + 5.58e9T^{2} \)
97 \( 1 - 8.27e4T + 8.58e9T^{2} \)
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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

−10.28494386615010272586590425685, −9.654104555525310686942746447211, −8.062531085122729615163538111837, −7.66439602550003136144894996742, −6.47689806603797957323805446308, −5.09206091609041731350937062248, −4.90325646291574798880288462185, −3.31759992455604673670436389590, −2.31906315476784917291311288244, −0.943552558015068996507270231004, 0.943552558015068996507270231004, 2.31906315476784917291311288244, 3.31759992455604673670436389590, 4.90325646291574798880288462185, 5.09206091609041731350937062248, 6.47689806603797957323805446308, 7.66439602550003136144894996742, 8.062531085122729615163538111837, 9.654104555525310686942746447211, 10.28494386615010272586590425685

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