Properties

Label 2-450-1.1-c5-0-11
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 + 17.4·7-s − 64·8-s + 645.·11-s + 1.09e3·13-s − 69.7·14-s + 256·16-s + 1.16e3·17-s − 2.24e3·19-s − 2.58e3·22-s + 500·23-s − 4.39e3·26-s + 278.·28-s − 470.·29-s + 3.85e3·31-s − 1.02e3·32-s − 4.66e3·34-s − 6.99e3·37-s + 8.97e3·38-s + 9.58e3·41-s + 5.30e3·43-s + 1.03e4·44-s − 2.00e3·46-s + 1.99e4·47-s − 1.65e4·49-s + 1.75e4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.134·7-s − 0.353·8-s + 1.60·11-s + 1.80·13-s − 0.0950·14-s + 0.250·16-s + 0.978·17-s − 1.42·19-s − 1.13·22-s + 0.197·23-s − 1.27·26-s + 0.0672·28-s − 0.103·29-s + 0.720·31-s − 0.176·32-s − 0.691·34-s − 0.839·37-s + 1.00·38-s + 0.890·41-s + 0.437·43-s + 0.803·44-s − 0.139·46-s + 1.31·47-s − 0.981·49-s + 0.901·52-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

\(L(3)\) \(\approx\) \(1.983217381\)
\(L(\frac12)\) \(\approx\) \(1.983217381\)
\(L(\frac{7}{2})\) 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 \( 1 - 17.4T + 1.68e4T^{2} \)
11 \( 1 - 645.T + 1.61e5T^{2} \)
13 \( 1 - 1.09e3T + 3.71e5T^{2} \)
17 \( 1 - 1.16e3T + 1.41e6T^{2} \)
19 \( 1 + 2.24e3T + 2.47e6T^{2} \)
23 \( 1 - 500T + 6.43e6T^{2} \)
29 \( 1 + 470.T + 2.05e7T^{2} \)
31 \( 1 - 3.85e3T + 2.86e7T^{2} \)
37 \( 1 + 6.99e3T + 6.93e7T^{2} \)
41 \( 1 - 9.58e3T + 1.15e8T^{2} \)
43 \( 1 - 5.30e3T + 1.47e8T^{2} \)
47 \( 1 - 1.99e4T + 2.29e8T^{2} \)
53 \( 1 - 1.14e3T + 4.18e8T^{2} \)
59 \( 1 + 3.73e4T + 7.14e8T^{2} \)
61 \( 1 + 3.81e4T + 8.44e8T^{2} \)
67 \( 1 - 3.62e4T + 1.35e9T^{2} \)
71 \( 1 + 1.66e4T + 1.80e9T^{2} \)
73 \( 1 + 6.93e4T + 2.07e9T^{2} \)
79 \( 1 + 2.06e4T + 3.07e9T^{2} \)
83 \( 1 - 9.69e4T + 3.93e9T^{2} \)
89 \( 1 - 5.96e4T + 5.58e9T^{2} \)
97 \( 1 + 5.82e3T + 8.58e9T^{2} \)
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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

−10.34142023066881220351787659828, −9.128307403849466888584079972775, −8.706333249986652334177457212592, −7.69160569065001587839842454140, −6.48166725768486418222701118376, −5.99161735059228648322496642937, −4.28581694719477274901734545696, −3.34908775522282582454415210467, −1.71425303598199202751094112356, −0.878558582788307588905801996738, 0.878558582788307588905801996738, 1.71425303598199202751094112356, 3.34908775522282582454415210467, 4.28581694719477274901734545696, 5.99161735059228648322496642937, 6.48166725768486418222701118376, 7.69160569065001587839842454140, 8.706333249986652334177457212592, 9.128307403849466888584079972775, 10.34142023066881220351787659828

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