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Properties

Label 2-450-25.21-c1-0-2
Degree $2$
Conductor $450$
Sign $-0.929 - 0.368i$
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 + (−0.690 + 2.12i)5-s + 0.381·7-s + (0.809 − 0.587i)8-s + (−1.80 − 1.31i)10-s + (−0.427 + 1.31i)11-s + (0.763 + 2.35i)13-s + (−0.118 + 0.363i)14-s + (0.309 + 0.951i)16-s + (−2.61 + 1.90i)17-s + (−6.23 + 4.53i)19-s + (1.80 − 1.31i)20-s + (−1.11 − 0.812i)22-s + (1.38 − 4.25i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.309 + 0.951i)5-s + 0.144·7-s + (0.286 − 0.207i)8-s + (−0.572 − 0.415i)10-s + (−0.128 + 0.396i)11-s + (0.211 + 0.652i)13-s + (−0.0315 + 0.0970i)14-s + (0.0772 + 0.237i)16-s + (−0.634 + 0.461i)17-s + (−1.43 + 1.03i)19-s + (0.404 − 0.293i)20-s + (−0.238 − 0.173i)22-s + (0.288 − 0.886i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.929 - 0.368i$
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: \((2,\ 450,\ (\ :1/2),\ -0.929 - 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.154684 + 0.810885i\)
\(L(\frac12)\) \(\approx\) \(0.154684 + 0.810885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.690 - 2.12i)T \)
good7 \( 1 - 0.381T + 7T^{2} \)
11 \( 1 + (0.427 - 1.31i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.763 - 2.35i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.61 - 1.90i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (6.23 - 4.53i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.38 + 4.25i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.381 + 0.277i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.54 - 2.57i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.47 - 7.60i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.38 - 7.33i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + (9.47 + 6.88i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-7.35 - 5.34i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.427 - 1.31i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.23 + 6.88i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.47 + 6.15i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-11.7 - 8.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.85 + 11.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.73 - 1.98i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.16 + 5.20i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.85 - 11.8i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-4.54 - 3.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.23407723062749635865439173088, −10.57720537410809473178135577811, −9.703169449207090117861449169070, −8.531217558786399080217374978604, −7.87196625262933071575018408316, −6.65616773504059258437548906526, −6.32540339936367648394975428010, −4.75368093632331611941730933131, −3.76352068747889627726116672067, −2.10684546670541035528084775205, 0.54536155604252720885828216010, 2.23113886391108802450760837228, 3.71573102516154497726879123359, 4.72212198527533301212670133830, 5.72218615148543923875935324776, 7.21763106303770352955924755672, 8.259043735279781319863364902360, 8.920504747843749917823702380663, 9.696173723399026573005223211468, 11.13253690131965515422719434011

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