Properties

Degree $2$
Conductor $450$
Sign $-0.728 - 0.684i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

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 − 3·7-s + (0.309 − 0.951i)8-s + (−0.690 + 2.12i)10-s + (−0.190 − 0.138i)11-s + (−0.809 + 0.587i)13-s + (2.42 + 1.76i)14-s + (−0.809 + 0.587i)16-s + (−2.42 + 7.46i)17-s + (−0.263 + 0.812i)19-s + (1.80 − 1.31i)20-s + (0.0729 + 0.224i)22-s + (−5.04 − 3.66i)23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.309 − 0.951i)5-s − 1.13·7-s + (0.109 − 0.336i)8-s + (−0.218 + 0.672i)10-s + (−0.0575 − 0.0418i)11-s + (−0.224 + 0.163i)13-s + (0.648 + 0.471i)14-s + (−0.202 + 0.146i)16-s + (−0.588 + 1.81i)17-s + (−0.0605 + 0.186i)19-s + (0.404 − 0.293i)20-s + (0.0155 + 0.0478i)22-s + (−1.05 − 0.764i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.728 - 0.684i$
Motivic weight: \(1\)
Character: $\chi_{450} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.728 - 0.684i)\)

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 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 \)
5 \( 1 + (0.690 + 2.12i)T \)
good7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + (0.190 + 0.138i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.42 - 7.46i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.263 - 0.812i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.04 + 3.66i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.163 - 0.502i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.30 + 4.02i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.92 - 4.30i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.04 - 4.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + (1.83 + 5.65i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.472 + 1.45i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.61 + 2.62i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.73 - 1.26i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.78 - 5.48i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.927 - 2.85i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.61 - 3.35i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.854 + 2.62i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.16 + 12.8i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (3.61 + 2.62i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.26 + 10.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.26328941547146377056358318445, −9.782259874174138474562083181236, −8.609467959933360660663665116311, −8.230378387629925224543962587662, −6.83841495490255345707359449930, −5.92830136798524843262160322659, −4.41762481619095627401582284407, −3.48093925008119709084154433439, −1.86160265433087180604029033674, 0, 2.49689770459919310804008872713, 3.58858543393454466736065080956, 5.17859677913348361385269910383, 6.42718425893663748905887828693, 7.00440801349771244598839350277, 7.82348501493584776535769314511, 9.100248729028358450552680113994, 9.795218599706978399167678000406, 10.54882286473582133547660390781

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