Properties

Degree $2$
Conductor $450$
Sign $0.960 - 0.276i$
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 + (−2.15 + 0.587i)5-s + 1.83·7-s + (0.809 + 0.587i)8-s + (1.22 + 1.87i)10-s + (0.566 + 1.74i)11-s + (−0.747 + 2.29i)13-s + (−0.566 − 1.74i)14-s + (0.309 − 0.951i)16-s + (2.25 + 1.63i)17-s + (1.35 + 0.982i)19-s + (1.39 − 1.74i)20-s + (1.48 − 1.07i)22-s + (2.39 + 7.38i)23-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.964 + 0.262i)5-s + 0.692·7-s + (0.286 + 0.207i)8-s + (0.387 + 0.591i)10-s + (0.170 + 0.525i)11-s + (−0.207 + 0.637i)13-s + (−0.151 − 0.466i)14-s + (0.0772 − 0.237i)16-s + (0.546 + 0.396i)17-s + (0.310 + 0.225i)19-s + (0.313 − 0.389i)20-s + (0.316 − 0.229i)22-s + (0.500 + 1.54i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.960 - 0.276i$
Motivic weight: \(1\)
Character: $\chi_{450} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.960 - 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03026 + 0.145530i\)
\(L(\frac12)\) \(\approx\) \(1.03026 + 0.145530i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.587i)T \)
good7 \( 1 - 1.83T + 7T^{2} \)
11 \( 1 + (-0.566 - 1.74i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.747 - 2.29i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.25 - 1.63i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.35 - 0.982i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.39 - 7.38i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.13 + 4.45i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.28 - 3.11i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.406 - 1.25i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.08 - 3.34i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.30T + 43T^{2} \)
47 \( 1 + (-1.48 + 1.07i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.27 - 3.83i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.79 + 8.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.799 - 2.46i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.68 + 5.58i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.247 + 0.179i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.61 - 14.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.79 + 2.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.15 + 3.74i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.02 - 3.15i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.97 - 6.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.34851450721119146067624753459, −10.30115627165611347361536954611, −9.483854359177224141180986753319, −8.324841270069983281565088255613, −7.71850291502998222364306494019, −6.68806322238356209360309212887, −5.05981850346342886113287762348, −4.16667921516593358745001975001, −3.06099321466238628544769289602, −1.46639640858584809221119684048, 0.810720701918057698259400249482, 3.08747181847508739622640076525, 4.49860733272101886444310352838, 5.22395907222611187426636863847, 6.53850159441841130442668064804, 7.52484121723798891972586971819, 8.279657901952141419186403258659, 8.867472570729703288573169100646, 10.16694791823116845312379986082, 11.01992510490007913408179426106

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