Properties

Label 2-450-1.1-c3-0-15
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 7-s − 8·8-s − 42·11-s + 67·13-s − 2·14-s + 16·16-s + 54·17-s − 115·19-s + 84·22-s − 162·23-s − 134·26-s + 4·28-s + 210·29-s − 193·31-s − 32·32-s − 108·34-s + 286·37-s + 230·38-s − 12·41-s − 263·43-s − 168·44-s + 324·46-s + 414·47-s − 342·49-s + 268·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.0539·7-s − 0.353·8-s − 1.15·11-s + 1.42·13-s − 0.0381·14-s + 1/4·16-s + 0.770·17-s − 1.38·19-s + 0.814·22-s − 1.46·23-s − 1.01·26-s + 0.0269·28-s + 1.34·29-s − 1.11·31-s − 0.176·32-s − 0.544·34-s + 1.27·37-s + 0.981·38-s − 0.0457·41-s − 0.932·43-s − 0.575·44-s + 1.03·46-s + 1.28·47-s − 0.997·49-s + 0.714·52-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: \(1\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - T + p^{3} T^{2} \)
11 \( 1 + 42 T + p^{3} T^{2} \)
13 \( 1 - 67 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 + 115 T + p^{3} T^{2} \)
23 \( 1 + 162 T + p^{3} T^{2} \)
29 \( 1 - 210 T + p^{3} T^{2} \)
31 \( 1 + 193 T + p^{3} T^{2} \)
37 \( 1 - 286 T + p^{3} T^{2} \)
41 \( 1 + 12 T + p^{3} T^{2} \)
43 \( 1 + 263 T + p^{3} T^{2} \)
47 \( 1 - 414 T + p^{3} T^{2} \)
53 \( 1 + 192 T + p^{3} T^{2} \)
59 \( 1 + 690 T + p^{3} T^{2} \)
61 \( 1 + 733 T + p^{3} T^{2} \)
67 \( 1 + 299 T + p^{3} T^{2} \)
71 \( 1 - 228 T + p^{3} T^{2} \)
73 \( 1 + 938 T + p^{3} T^{2} \)
79 \( 1 + 160 T + p^{3} T^{2} \)
83 \( 1 + 462 T + p^{3} T^{2} \)
89 \( 1 - 240 T + p^{3} T^{2} \)
97 \( 1 - 511 T + p^{3} T^{2} \)
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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

−10.42742159732335419909950751930, −9.285560660293003633864848810905, −8.261227152270461626443946015735, −7.83585590572544441062544354583, −6.45215667214553317431699458332, −5.72176214258874655195980911260, −4.25678487168865724890900424863, −2.92238207896602842104644539769, −1.57819120348360981361813177911, 0, 1.57819120348360981361813177911, 2.92238207896602842104644539769, 4.25678487168865724890900424863, 5.72176214258874655195980911260, 6.45215667214553317431699458332, 7.83585590572544441062544354583, 8.261227152270461626443946015735, 9.285560660293003633864848810905, 10.42742159732335419909950751930

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