Properties

Label 2-450-1.1-c3-0-1
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 − 14·7-s − 8·8-s − 6·11-s − 68·13-s + 28·14-s + 16·16-s + 78·17-s + 44·19-s + 12·22-s + 120·23-s + 136·26-s − 56·28-s − 126·29-s − 244·31-s − 32·32-s − 156·34-s + 304·37-s − 88·38-s + 480·41-s − 104·43-s − 24·44-s − 240·46-s + 600·47-s − 147·49-s − 272·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.164·11-s − 1.45·13-s + 0.534·14-s + 1/4·16-s + 1.11·17-s + 0.531·19-s + 0.116·22-s + 1.08·23-s + 1.02·26-s − 0.377·28-s − 0.806·29-s − 1.41·31-s − 0.176·32-s − 0.786·34-s + 1.35·37-s − 0.375·38-s + 1.82·41-s − 0.368·43-s − 0.0822·44-s − 0.769·46-s + 1.86·47-s − 3/7·49-s − 0.725·52-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

\(L(2)\) \(\approx\) \(1.043471466\)
\(L(\frac12)\) \(\approx\) \(1.043471466\)
\(L(\frac{5}{2})\) 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 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 + 6 T + p^{3} T^{2} \)
13 \( 1 + 68 T + p^{3} T^{2} \)
17 \( 1 - 78 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 + 126 T + p^{3} T^{2} \)
31 \( 1 + 244 T + p^{3} T^{2} \)
37 \( 1 - 304 T + p^{3} T^{2} \)
41 \( 1 - 480 T + p^{3} T^{2} \)
43 \( 1 + 104 T + p^{3} T^{2} \)
47 \( 1 - 600 T + p^{3} T^{2} \)
53 \( 1 + 258 T + p^{3} T^{2} \)
59 \( 1 + 534 T + p^{3} T^{2} \)
61 \( 1 - 362 T + p^{3} T^{2} \)
67 \( 1 - 4 p T + p^{3} T^{2} \)
71 \( 1 - 972 T + p^{3} T^{2} \)
73 \( 1 + 470 T + p^{3} T^{2} \)
79 \( 1 - 1244 T + p^{3} T^{2} \)
83 \( 1 - 396 T + p^{3} T^{2} \)
89 \( 1 - 972 T + p^{3} T^{2} \)
97 \( 1 - 46 T + p^{3} T^{2} \)
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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

−10.52873448966348334593758137999, −9.527384514854369337038634369028, −9.263081368238322834588518547265, −7.69430038399690922992765784190, −7.33122259617803236105047640323, −6.07154081068786469847128675451, −5.06500632155895097057536809007, −3.47009662258862233352870022221, −2.39179668370222691032868037628, −0.70456975136585232026115801955, 0.70456975136585232026115801955, 2.39179668370222691032868037628, 3.47009662258862233352870022221, 5.06500632155895097057536809007, 6.07154081068786469847128675451, 7.33122259617803236105047640323, 7.69430038399690922992765784190, 9.263081368238322834588518547265, 9.527384514854369337038634369028, 10.52873448966348334593758137999

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