Properties

Degree $2$
Conductor $450$
Sign $0.636 + 0.771i$
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 + (2.02 − 0.951i)5-s + 3.77·7-s + (−0.309 − 0.951i)8-s + (1.07 − 1.95i)10-s + (−3.05 + 2.21i)11-s + (−2.56 − 1.86i)13-s + (3.05 − 2.21i)14-s + (−0.809 − 0.587i)16-s + (0.430 + 1.32i)17-s + (1.20 + 3.72i)19-s + (−0.279 − 2.21i)20-s + (−1.16 + 3.58i)22-s + (0.720 − 0.523i)23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.905 − 0.425i)5-s + 1.42·7-s + (−0.109 − 0.336i)8-s + (0.340 − 0.619i)10-s + (−0.920 + 0.668i)11-s + (−0.712 − 0.517i)13-s + (0.816 − 0.592i)14-s + (−0.202 − 0.146i)16-s + (0.104 + 0.321i)17-s + (0.277 + 0.853i)19-s + (−0.0624 − 0.496i)20-s + (−0.248 + 0.765i)22-s + (0.150 − 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.636 + 0.771i$
Motivic weight: \(1\)
Character: $\chi_{450} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.636 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09678 - 0.987958i\)
\(L(\frac12)\) \(\approx\) \(2.09678 - 0.987958i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 0.951i)T \)
good7 \( 1 - 3.77T + 7T^{2} \)
11 \( 1 + (3.05 - 2.21i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.56 + 1.86i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.430 - 1.32i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.20 - 3.72i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.720 + 0.523i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.0152 - 0.0468i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.72 + 5.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.70 + 4.14i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.20 + 0.875i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.69T + 43T^{2} \)
47 \( 1 + (1.16 - 3.58i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.58 - 11.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.558 + 0.405i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.38 - 6.08i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.73 - 14.5i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.06 + 6.36i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.18 - 3.03i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.558 - 1.71i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.08 - 9.47i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.7 + 8.52i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.0278 - 0.0857i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.85323658544293765542849335972, −10.28893561850749627512963919252, −9.370887921141188496289747615361, −8.169757562543099033392265656728, −7.37479680591133865416342545137, −5.78985093496032736759548337778, −5.19147893142958680608662909544, −4.33888315945612470963091240551, −2.55714357764392171769783156931, −1.57351132688070417888829746402, 1.95368324158015536629900332659, 3.13541108751778977217213067803, 4.95073826756198367546128255647, 5.19785598001652967991005606492, 6.53209068885932105836208812156, 7.42918751408149043500280998632, 8.341623189707837390887934542348, 9.342326090366786326984383804464, 10.54237208538192657119467919167, 11.19929387020706111160429175357

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