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Properties

Label 2-450-25.21-c1-0-5
Degree $2$
Conductor $450$
Sign $0.367 - 0.929i$
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 + (1.01 + 1.99i)5-s + 4.77·7-s + (0.809 − 0.587i)8-s + (−2.20 + 0.350i)10-s + (0.0788 − 0.242i)11-s + (−1.32 − 4.07i)13-s + (−1.47 + 4.54i)14-s + (0.309 + 0.951i)16-s + (1.88 − 1.37i)17-s + (1.12 − 0.814i)19-s + (0.349 − 2.20i)20-s + (0.206 + 0.149i)22-s + (−2.31 + 7.11i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.454 + 0.890i)5-s + 1.80·7-s + (0.286 − 0.207i)8-s + (−0.698 + 0.110i)10-s + (0.0237 − 0.0731i)11-s + (−0.367 − 1.13i)13-s + (−0.394 + 1.21i)14-s + (0.0772 + 0.237i)16-s + (0.457 − 0.332i)17-s + (0.257 − 0.186i)19-s + (0.0781 − 0.493i)20-s + (0.0440 + 0.0319i)22-s + (−0.481 + 1.48i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.367 - 0.929i$
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: \((2,\ 450,\ (\ :1/2),\ 0.367 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27035 + 0.863581i\)
\(L(\frac12)\) \(\approx\) \(1.27035 + 0.863581i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.01 - 1.99i)T \)
good7 \( 1 - 4.77T + 7T^{2} \)
11 \( 1 + (-0.0788 + 0.242i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.32 + 4.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.88 + 1.37i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.12 + 0.814i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.31 - 7.11i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.13 - 3.73i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.143 - 0.103i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.57 - 4.84i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.681 - 2.09i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.60T + 43T^{2} \)
47 \( 1 + (6.09 + 4.42i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (10.9 + 7.93i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.20 + 6.78i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.67 + 11.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (6.44 - 4.68i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (8.27 + 6.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.98 - 9.18i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.91 - 2.11i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.8 + 9.36i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.45 - 10.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-12.7 - 9.26i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.15841787921002990499677225491, −10.30879861029389171220993274092, −9.514122147539971015467040397375, −8.131395227973636004014094201283, −7.78641233397262130189560609266, −6.72622007234948557269614553256, −5.49737696689300702920770475696, −4.91608097993058217218818021015, −3.21840542793455438155724827402, −1.60584749158252476498422702772, 1.32288109552870374377471159103, 2.27048368821102107218972223964, 4.35545423252572581997464735860, 4.75063638258759717445338102085, 6.02714515046612583391822154909, 7.61719526351811018242849304452, 8.382088176693146439667439646976, 9.074693567530839020359396312819, 10.09108910203018932582030461427, 10.93839736617183626728945263359

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