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Properties

Label 2-450-25.16-c1-0-12
Degree $2$
Conductor $450$
Sign $-0.713 + 0.700i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−1.97 − 1.05i)5-s − 5.04·7-s + (−0.309 + 0.951i)8-s + (−0.974 − 2.01i)10-s + (−4.35 − 3.16i)11-s + (2.78 − 2.02i)13-s + (−4.08 − 2.96i)14-s + (−0.809 + 0.587i)16-s + (1.15 − 3.54i)17-s + (−1.51 + 4.65i)19-s + (0.394 − 2.20i)20-s + (−1.66 − 5.11i)22-s + (−1.82 − 1.32i)23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.881 − 0.472i)5-s − 1.90·7-s + (−0.109 + 0.336i)8-s + (−0.308 − 0.636i)10-s + (−1.31 − 0.953i)11-s + (0.771 − 0.560i)13-s + (−1.09 − 0.793i)14-s + (−0.202 + 0.146i)16-s + (0.279 − 0.860i)17-s + (−0.346 + 1.06i)19-s + (0.0882 − 0.492i)20-s + (−0.354 − 1.09i)22-s + (−0.381 − 0.276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.713 + 0.700i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106492 - 0.260491i\)
\(L(\frac12)\) \(\approx\) \(0.106492 - 0.260491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 \)
5 \( 1 + (1.97 + 1.05i)T \)
good7 \( 1 + 5.04T + 7T^{2} \)
11 \( 1 + (4.35 + 3.16i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.78 + 2.02i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.15 + 3.54i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.51 - 4.65i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.82 + 1.32i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.153 - 0.473i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.05 - 6.33i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.558 - 0.405i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.12 - 6.62i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.179T + 43T^{2} \)
47 \( 1 + (3.34 + 10.3i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.526 + 1.61i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.45 + 3.24i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.55 + 6.21i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.86 + 8.81i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.15 - 3.55i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.92 + 2.85i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.52 + 7.77i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.72 - 11.4i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.89 + 1.37i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.584 - 1.79i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.75931878108997388887334046860, −9.930583770534091390086494969726, −8.656305684193389812613414387872, −8.027052876639246931665847847264, −6.91848815021701207483722225508, −5.98666916235088653966824729458, −5.10328967207628608419352687534, −3.53213559253872830025146858147, −3.16186298749283095140170441485, −0.13559570279173585819238669747, 2.50298484163842836461536948158, 3.48601004136572096760865598028, 4.35560555912991559672898127241, 5.84460923913963787292960242142, 6.71893973384638777231455616523, 7.52597766333014282555791020771, 8.864577389108717982878171519597, 9.959999528769004827539818666344, 10.51255935036581702693501632294, 11.47236509450626345495474078159

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