Properties

Label 2-450-1.1-c5-0-32
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

Related objects

Downloads

Learn more about

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: $\chi_{450} (1, \cdot )$
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 \( 1 + 141.T + 1.68e4T^{2} \)
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 \( 1 - 4.35e4T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.935705106400373296396112955645, −9.061845747405042369247456006498, −7.80403298693254137593871834033, −6.85952698948006087644751169679, −6.02223665843315747869417568295, −5.09052714066856837845047209432, −3.75222339248455484122105984664, −3.07709350036095657466241134733, −1.61480369758448540050699361278, 0, 1.61480369758448540050699361278, 3.07709350036095657466241134733, 3.75222339248455484122105984664, 5.09052714066856837845047209432, 6.02223665843315747869417568295, 6.85952698948006087644751169679, 7.80403298693254137593871834033, 9.061845747405042369247456006498, 9.935705106400373296396112955645

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