Properties

Label 2-450-1.1-c5-0-33
Degree $2$
Conductor $450$
Sign $-1$
Analytic cond. $72.1727$
Root an. cond. $8.49545$
Motivic weight $5$
Arithmetic yes
Rational yes
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 − 47·7-s + 64·8-s − 222·11-s − 101·13-s − 188·14-s + 256·16-s + 162·17-s + 1.68e3·19-s − 888·22-s + 306·23-s − 404·26-s − 752·28-s − 7.89e3·29-s − 8.59e3·31-s + 1.02e3·32-s + 648·34-s − 8.64e3·37-s + 6.74e3·38-s + 1.81e4·41-s − 1.43e4·43-s − 3.55e3·44-s + 1.22e3·46-s − 1.09e3·47-s − 1.45e4·49-s − 1.61e3·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.362·7-s + 0.353·8-s − 0.553·11-s − 0.165·13-s − 0.256·14-s + 1/4·16-s + 0.135·17-s + 1.07·19-s − 0.391·22-s + 0.120·23-s − 0.117·26-s − 0.181·28-s − 1.74·29-s − 1.60·31-s + 0.176·32-s + 0.0961·34-s − 1.03·37-s + 0.757·38-s + 1.68·41-s − 1.18·43-s − 0.276·44-s + 0.0852·46-s − 0.0725·47-s − 0.868·49-s − 0.0828·52-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: yes
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 \( 1 - p^{2} T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 47 T + p^{5} T^{2} \)
11 \( 1 + 222 T + p^{5} T^{2} \)
13 \( 1 + 101 T + p^{5} T^{2} \)
17 \( 1 - 162 T + p^{5} T^{2} \)
19 \( 1 - 1685 T + p^{5} T^{2} \)
23 \( 1 - 306 T + p^{5} T^{2} \)
29 \( 1 + 7890 T + p^{5} T^{2} \)
31 \( 1 + 8593 T + p^{5} T^{2} \)
37 \( 1 + 8642 T + p^{5} T^{2} \)
41 \( 1 - 18168 T + p^{5} T^{2} \)
43 \( 1 + 14351 T + p^{5} T^{2} \)
47 \( 1 + 1098 T + p^{5} T^{2} \)
53 \( 1 - 17916 T + p^{5} T^{2} \)
59 \( 1 + 17610 T + p^{5} T^{2} \)
61 \( 1 + 21853 T + p^{5} T^{2} \)
67 \( 1 + 107 T + p^{5} T^{2} \)
71 \( 1 - 40728 T + p^{5} T^{2} \)
73 \( 1 + 34706 T + p^{5} T^{2} \)
79 \( 1 + 69160 T + p^{5} T^{2} \)
83 \( 1 + 108534 T + p^{5} T^{2} \)
89 \( 1 + 35040 T + p^{5} T^{2} \)
97 \( 1 - 823 T + p^{5} T^{2} \)
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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

−9.902447220276869551716774462002, −9.039563348056528658256834650775, −7.70850919948912657872804950326, −7.07943981898390846863713326420, −5.81199614899744897054865255670, −5.15980766807950767393081042988, −3.85905229202199315357388561496, −2.95200846062194049193019151785, −1.66145096760805090630678750778, 0, 1.66145096760805090630678750778, 2.95200846062194049193019151785, 3.85905229202199315357388561496, 5.15980766807950767393081042988, 5.81199614899744897054865255670, 7.07943981898390846863713326420, 7.70850919948912657872804950326, 9.039563348056528658256834650775, 9.902447220276869551716774462002

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