Properties

Degree $2$
Conductor $450$
Sign $-0.0627 + 0.998i$
Motivic weight $1$
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 + (−1.80 + 1.31i)5-s + 2·7-s + (−0.809 + 0.587i)8-s + (0.690 + 2.12i)10-s + (1.61 − 4.97i)11-s + (−1.5 − 4.61i)13-s + (0.618 − 1.90i)14-s + (0.309 + 0.951i)16-s + (6.35 − 4.61i)17-s + (2.23 − 1.62i)19-s + 2.23·20-s + (−4.23 − 3.07i)22-s + (−1.85 + 5.70i)23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.809 + 0.587i)5-s + 0.755·7-s + (−0.286 + 0.207i)8-s + (0.218 + 0.672i)10-s + (0.487 − 1.50i)11-s + (−0.416 − 1.28i)13-s + (0.165 − 0.508i)14-s + (0.0772 + 0.237i)16-s + (1.54 − 1.11i)17-s + (0.512 − 0.372i)19-s + 0.499·20-s + (−0.903 − 0.656i)22-s + (−0.386 + 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0627 + 0.998i$
Motivic weight: \(1\)
Character: $\chi_{450} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.0627 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934932 - 0.995601i\)
\(L(\frac12)\) \(\approx\) \(0.934932 - 0.995601i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (-1.61 + 4.97i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.5 + 4.61i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-6.35 + 4.61i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.23 + 1.62i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.85 - 5.70i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.11 + 0.812i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3 - 2.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.663 + 2.04i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.88 - 5.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + (-3.85 - 2.80i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.92 - 5.03i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.76 + 8.50i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.85 + 5.70i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (11.4 + 8.33i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.972 - 2.99i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.85 + 3.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.427 - 1.31i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-11.2 - 8.14i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.08646484384717044660128577714, −10.20715186225132252860089643556, −9.162150856189590915528088285590, −7.977694205447265737245026042672, −7.48973596222358083568363141569, −5.86524893538001477862829959635, −5.04594811785913824421766271216, −3.56249681580468077846087389582, −2.97685753764307746585473005076, −0.896885527656177482216634935856, 1.70239122074357906676292239358, 3.89923457097070615483004516330, 4.51231501264071917048327503538, 5.51902732323806502111097695655, 6.90016401126305450011119505837, 7.61423358116035931439588100382, 8.404217952261960755601023260270, 9.349789228608469398237930124075, 10.29998337577378028743722663393, 11.71801849093352782563306979879

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