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Properties

Label 2-450-25.21-c1-0-1
Degree $2$
Conductor $450$
Sign $-0.721 - 0.692i$
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 + (−1.95 − 1.08i)5-s + 0.757·7-s + (0.809 − 0.587i)8-s + (1.63 − 1.52i)10-s + (−1.05 + 3.24i)11-s + (1.64 + 5.06i)13-s + (−0.233 + 0.720i)14-s + (0.309 + 0.951i)16-s + (−3.87 + 2.81i)17-s + (−0.298 + 0.216i)19-s + (0.941 + 2.02i)20-s + (−2.76 − 2.00i)22-s + (−1.43 + 4.41i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.873 − 0.486i)5-s + 0.286·7-s + (0.286 − 0.207i)8-s + (0.517 − 0.481i)10-s + (−0.318 + 0.979i)11-s + (0.456 + 1.40i)13-s + (−0.0625 + 0.192i)14-s + (0.0772 + 0.237i)16-s + (−0.939 + 0.682i)17-s + (−0.0683 + 0.0496i)19-s + (0.210 + 0.453i)20-s + (−0.589 − 0.427i)22-s + (−0.298 + 0.920i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.721 - 0.692i$
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: \((2,\ 450,\ (\ :1/2),\ -0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.271256 + 0.674773i\)
\(L(\frac12)\) \(\approx\) \(0.271256 + 0.674773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.95 + 1.08i)T \)
good7 \( 1 - 0.757T + 7T^{2} \)
11 \( 1 + (1.05 - 3.24i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.64 - 5.06i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.87 - 2.81i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.298 - 0.216i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.43 - 4.41i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.127 + 0.0922i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.70 - 2.69i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.49 + 7.68i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.678 + 2.08i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.65T + 43T^{2} \)
47 \( 1 + (-9.14 - 6.64i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.41 - 3.20i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.79 + 8.60i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.05 - 12.4i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.28 + 1.65i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.67 + 4.12i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.80 - 11.6i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (12.2 + 8.91i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.60 + 4.07i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.281 + 0.865i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.80 - 2.03i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.39287209935754563606491244863, −10.56742569016983165963883565668, −9.219518680595523183507949352165, −8.795325624871152545673438922999, −7.63114193593920170675299862676, −7.07699540538675988134521129394, −5.83924689696652829052650448374, −4.59313956791634467362375639584, −3.97502790473067388622020670349, −1.77149341301808118654989274582, 0.49502011753690716239847479961, 2.63069375563650797683522125070, 3.54001758858932607515851767757, 4.71890570656332207853253130771, 5.99798393384330217736946035237, 7.29373742071066246413132182366, 8.215165221356931530513032000804, 8.771682008966636786865667022335, 10.17925858329448248118344713792, 10.88534477916814770447914282723

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