Properties

Label 2-450-1.1-c5-0-30
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 + 233.·7-s − 64·8-s + 89.7·11-s − 209.·13-s − 934.·14-s + 256·16-s − 226.·17-s − 2.18e3·19-s − 358.·22-s − 4.29e3·23-s + 836.·26-s + 3.73e3·28-s − 3.55e3·29-s + 6.90e3·31-s − 1.02e3·32-s + 907.·34-s − 5.43e3·37-s + 8.72e3·38-s − 6.48e3·41-s − 2.19e4·43-s + 1.43e3·44-s + 1.71e4·46-s + 7.71e3·47-s + 3.78e4·49-s − 3.34e3·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.80·7-s − 0.353·8-s + 0.223·11-s − 0.343·13-s − 1.27·14-s + 0.250·16-s − 0.190·17-s − 1.38·19-s − 0.158·22-s − 1.69·23-s + 0.242·26-s + 0.901·28-s − 0.785·29-s + 1.28·31-s − 0.176·32-s + 0.134·34-s − 0.652·37-s + 0.979·38-s − 0.602·41-s − 1.81·43-s + 0.111·44-s + 1.19·46-s + 0.509·47-s + 2.24·49-s − 0.171·52-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 \( 1 - 233.T + 1.68e4T^{2} \)
11 \( 1 - 89.7T + 1.61e5T^{2} \)
13 \( 1 + 209.T + 3.71e5T^{2} \)
17 \( 1 + 226.T + 1.41e6T^{2} \)
19 \( 1 + 2.18e3T + 2.47e6T^{2} \)
23 \( 1 + 4.29e3T + 6.43e6T^{2} \)
29 \( 1 + 3.55e3T + 2.05e7T^{2} \)
31 \( 1 - 6.90e3T + 2.86e7T^{2} \)
37 \( 1 + 5.43e3T + 6.93e7T^{2} \)
41 \( 1 + 6.48e3T + 1.15e8T^{2} \)
43 \( 1 + 2.19e4T + 1.47e8T^{2} \)
47 \( 1 - 7.71e3T + 2.29e8T^{2} \)
53 \( 1 + 1.27e4T + 4.18e8T^{2} \)
59 \( 1 - 4.45e4T + 7.14e8T^{2} \)
61 \( 1 - 1.15e4T + 8.44e8T^{2} \)
67 \( 1 + 3.96e3T + 1.35e9T^{2} \)
71 \( 1 + 4.64e4T + 1.80e9T^{2} \)
73 \( 1 + 6.15e4T + 2.07e9T^{2} \)
79 \( 1 + 2.78e4T + 3.07e9T^{2} \)
83 \( 1 - 5.09e4T + 3.93e9T^{2} \)
89 \( 1 + 5.13e3T + 5.58e9T^{2} \)
97 \( 1 + 8.84e4T + 8.58e9T^{2} \)
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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

−9.961897107684374835223184260867, −8.598986492867825712336086994691, −8.269197551004626213009340533446, −7.30147171637064655759119770898, −6.19076695435597970548512202407, −5.00575888515818835319304954151, −4.03904863230108983270368989017, −2.25009107965815245076399840781, −1.51924495995894136039145930498, 0, 1.51924495995894136039145930498, 2.25009107965815245076399840781, 4.03904863230108983270368989017, 5.00575888515818835319304954151, 6.19076695435597970548512202407, 7.30147171637064655759119770898, 8.269197551004626213009340533446, 8.598986492867825712336086994691, 9.961897107684374835223184260867

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