Properties

Degree $2$
Conductor $450$
Sign $-0.535 - 0.844i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−1.80 − 1.31i)5-s − 3·7-s + (−0.809 + 0.587i)8-s + (−1.80 + 1.31i)10-s + (−1.30 + 4.02i)11-s + (0.309 + 0.951i)13-s + (−0.927 + 2.85i)14-s + (0.309 + 0.951i)16-s + (0.927 − 0.673i)17-s + (−4.73 + 3.44i)19-s + (0.690 + 2.12i)20-s + (3.42 + 2.48i)22-s + (0.545 − 1.67i)23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.809 − 0.587i)5-s − 1.13·7-s + (−0.286 + 0.207i)8-s + (−0.572 + 0.415i)10-s + (−0.394 + 1.21i)11-s + (0.0857 + 0.263i)13-s + (−0.247 + 0.762i)14-s + (0.0772 + 0.237i)16-s + (0.224 − 0.163i)17-s + (−1.08 + 0.789i)19-s + (0.154 + 0.475i)20-s + (0.730 + 0.530i)22-s + (0.113 − 0.349i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.535 - 0.844i$
Motivic weight: \(1\)
Character: $\chi_{450} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.31i)T \)
good7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + (1.30 - 4.02i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.927 + 0.673i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.545 + 1.67i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (7.66 + 5.56i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.190 + 0.138i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.57 + 7.91i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.454 + 1.40i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + (9.66 + 7.02i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-8.47 - 6.15i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.38 - 4.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.28 + 6.01i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.42 + 1.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.33i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.85 - 4.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.66 - 2.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.38 - 4.25i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (7.73 + 5.62i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.45628613905644049944504981219, −9.723841942380523264571083336675, −8.890526933757651532436505833656, −7.80603466757305555041438715132, −6.78583686158399291084737199917, −5.52780472724710961198477659242, −4.34500215480399794681566777645, −3.58751192218407239680299447081, −2.07865427878540105637347397105, 0, 3.05000763854884951145394741735, 3.67781629527325832773666766864, 5.14148103799231013348664421939, 6.31584176006740511372226371408, 6.88384144510362469895364221631, 8.014113829548500310170152011149, 8.717598904629978589190046805794, 9.865085377764723321942026647664, 10.86161890619523920080965326172

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