Properties

Degree $2$
Conductor $450$
Sign $0.425 + 0.904i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.690 − 2.12i)5-s + 2·7-s + (0.309 + 0.951i)8-s + (1.80 + 1.31i)10-s + (−0.618 + 0.449i)11-s + (−1.5 − 1.08i)13-s + (−1.61 + 1.17i)14-s + (−0.809 − 0.587i)16-s + (−0.354 − 1.08i)17-s + (−2.23 − 6.88i)19-s − 2.23·20-s + (0.236 − 0.726i)22-s + (4.85 − 3.52i)23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.309 − 0.951i)5-s + 0.755·7-s + (0.109 + 0.336i)8-s + (0.572 + 0.415i)10-s + (−0.186 + 0.135i)11-s + (−0.416 − 0.302i)13-s + (−0.432 + 0.314i)14-s + (−0.202 − 0.146i)16-s + (−0.0858 − 0.264i)17-s + (−0.512 − 1.57i)19-s − 0.499·20-s + (0.0503 − 0.154i)22-s + (1.01 − 0.735i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.425 + 0.904i$
Motivic weight: \(1\)
Character: $\chi_{450} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780242 - 0.495156i\)
\(L(\frac12)\) \(\approx\) \(0.780242 - 0.495156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 \)
5 \( 1 + (0.690 + 2.12i)T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (0.618 - 0.449i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.5 + 1.08i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.354 + 1.08i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.23 + 6.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.85 + 3.52i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.11 + 3.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3 + 9.23i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.16 - 5.20i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.11 - 2.99i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + (2.85 - 8.78i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.57 + 10.9i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.23 + 5.25i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.73 + 1.26i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.14 - 3.52i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.52 - 7.77i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.97 + 5.79i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.85 + 5.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.92 + 2.12i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.20 - 6.79i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.08942766241761525261418990750, −9.746591225036655107870723271901, −9.064711583429396716398569554792, −8.149813725394155248696895352027, −7.54302845413740753981934791856, −6.33696216486950193548422891042, −5.04210035629149781568732689764, −4.48712796213655218263172040301, −2.43438669948652211811061482876, −0.71984320839594658026703611666, 1.73386855585515372242556521952, 3.07079600558539847574732498628, 4.19216809407218172486966430316, 5.62999756074611182535050525585, 6.91638770031444455740452687596, 7.66294323268935259553996609597, 8.508132882155446432832783555405, 9.546618030015992556762025625345, 10.67754204950250129442404415587, 10.90382169797230282180071561683

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