Properties

Label 2-450-1.1-c3-0-12
Degree $2$
Conductor $450$
Sign $1$
Analytic cond. $26.5508$
Root an. cond. $5.15275$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 34·7-s + 8·8-s − 27·11-s + 28·13-s + 68·14-s + 16·16-s + 21·17-s + 35·19-s − 54·22-s − 78·23-s + 56·26-s + 136·28-s + 120·29-s + 182·31-s + 32·32-s + 42·34-s − 146·37-s + 70·38-s − 357·41-s + 148·43-s − 108·44-s − 156·46-s − 84·47-s + 813·49-s + 112·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.83·7-s + 0.353·8-s − 0.740·11-s + 0.597·13-s + 1.29·14-s + 1/4·16-s + 0.299·17-s + 0.422·19-s − 0.523·22-s − 0.707·23-s + 0.422·26-s + 0.917·28-s + 0.768·29-s + 1.05·31-s + 0.176·32-s + 0.211·34-s − 0.648·37-s + 0.298·38-s − 1.35·41-s + 0.524·43-s − 0.370·44-s − 0.500·46-s − 0.260·47-s + 2.37·49-s + 0.298·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.789221752\)
\(L(\frac12)\) \(\approx\) \(3.789221752\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 34 T + p^{3} T^{2} \)
11 \( 1 + 27 T + p^{3} T^{2} \)
13 \( 1 - 28 T + p^{3} T^{2} \)
17 \( 1 - 21 T + p^{3} T^{2} \)
19 \( 1 - 35 T + p^{3} T^{2} \)
23 \( 1 + 78 T + p^{3} T^{2} \)
29 \( 1 - 120 T + p^{3} T^{2} \)
31 \( 1 - 182 T + p^{3} T^{2} \)
37 \( 1 + 146 T + p^{3} T^{2} \)
41 \( 1 + 357 T + p^{3} T^{2} \)
43 \( 1 - 148 T + p^{3} T^{2} \)
47 \( 1 + 84 T + p^{3} T^{2} \)
53 \( 1 - 702 T + p^{3} T^{2} \)
59 \( 1 - 840 T + p^{3} T^{2} \)
61 \( 1 + 238 T + p^{3} T^{2} \)
67 \( 1 + 461 T + p^{3} T^{2} \)
71 \( 1 - 708 T + p^{3} T^{2} \)
73 \( 1 - 133 T + p^{3} T^{2} \)
79 \( 1 - 650 T + p^{3} T^{2} \)
83 \( 1 + 903 T + p^{3} T^{2} \)
89 \( 1 + 735 T + p^{3} T^{2} \)
97 \( 1 + 1106 T + p^{3} T^{2} \)
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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

−10.82880279180311744337821044755, −10.09815539209397703879692753129, −8.479116722888614261657466769868, −8.025444849379659780435528772547, −6.95109278803534259295523762520, −5.61833893429475927673655583965, −4.96368224057628740978331491419, −3.95125658356847747173826976507, −2.48230192572408641469503917246, −1.26850192288044301086170073447, 1.26850192288044301086170073447, 2.48230192572408641469503917246, 3.95125658356847747173826976507, 4.96368224057628740978331491419, 5.61833893429475927673655583965, 6.95109278803534259295523762520, 8.025444849379659780435528772547, 8.479116722888614261657466769868, 10.09815539209397703879692753129, 10.82880279180311744337821044755

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