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Properties

Label 2-450-25.21-c1-0-4
Degree $2$
Conductor $450$
Sign $0.708 - 0.705i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (2.15 − 0.587i)5-s − 0.833·7-s + (0.809 − 0.587i)8-s + (−0.107 + 2.23i)10-s + (−0.257 + 0.792i)11-s + (1.41 + 4.34i)13-s + (0.257 − 0.792i)14-s + (0.309 + 0.951i)16-s + (4.41 − 3.20i)17-s + (7.00 − 5.08i)19-s + (−2.09 − 0.792i)20-s + (−0.674 − 0.489i)22-s + (−1.09 + 3.35i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.964 − 0.262i)5-s − 0.314·7-s + (0.286 − 0.207i)8-s + (−0.0340 + 0.706i)10-s + (−0.0776 + 0.238i)11-s + (0.391 + 1.20i)13-s + (0.0688 − 0.211i)14-s + (0.0772 + 0.237i)16-s + (1.06 − 0.777i)17-s + (1.60 − 1.16i)19-s + (−0.467 − 0.177i)20-s + (−0.143 − 0.104i)22-s + (−0.227 + 0.700i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.708 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32738 + 0.548198i\)
\(L(\frac12)\) \(\approx\) \(1.32738 + 0.548198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 \)
5 \( 1 + (-2.15 + 0.587i)T \)
good7 \( 1 + 0.833T + 7T^{2} \)
11 \( 1 + (0.257 - 0.792i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.41 - 4.34i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.41 + 3.20i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-7.00 + 5.08i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.09 - 3.35i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.64 - 1.92i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.85 - 3.52i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.26 - 6.95i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.576 + 1.77i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 + (0.674 + 0.489i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.19 - 3.77i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.18 + 12.8i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.81 + 5.59i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.21 - 0.881i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (1.91 + 1.38i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.02 - 3.16i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.18 + 3.03i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.97 - 7.25i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.16 + 6.66i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (8.97 + 6.51i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.16332769416116875244355811864, −9.746042573534892328205627887743, −9.582847819623408303450062522764, −8.606773548918154139000142187388, −7.33084620767625068753593582233, −6.65012767468318600114230284680, −5.53245503363840911367167982868, −4.80854615056743770905059815945, −3.14320449872850306700899907657, −1.39522249770388458076991751870, 1.27187375771751555041006509621, 2.79664932788655801474686401926, 3.71246174956216500661705304440, 5.47428193446121688326208762948, 5.98181571991106552228304897673, 7.50527844083091780376952741376, 8.345017310705890849074187420962, 9.507726837457905512487557215606, 10.15232584462960545375175822246, 10.69469469394733914340946345595

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