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Properties

Label 2-450-25.6-c1-0-1
Degree $2$
Conductor $450$
Sign $-0.632 - 0.774i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
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 + (2.12 + 0.697i)5-s − 4.25·7-s + (−0.809 − 0.587i)8-s + (−0.00655 + 2.23i)10-s + (1.51 + 4.64i)11-s + (−1.43 + 4.41i)13-s + (−1.31 − 4.04i)14-s + (0.309 − 0.951i)16-s + (−0.815 − 0.592i)17-s + (1 + 0.726i)19-s + (−2.12 + 0.684i)20-s + (−3.95 + 2.87i)22-s + (1.38 + 4.25i)23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.950 + 0.311i)5-s − 1.60·7-s + (−0.286 − 0.207i)8-s + (−0.00207 + 0.707i)10-s + (0.455 + 1.40i)11-s + (−0.397 + 1.22i)13-s + (−0.351 − 1.08i)14-s + (0.0772 − 0.237i)16-s + (−0.197 − 0.143i)17-s + (0.229 + 0.166i)19-s + (−0.475 + 0.153i)20-s + (−0.843 + 0.612i)22-s + (0.288 + 0.886i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567102 + 1.19603i\)
\(L(\frac12)\) \(\approx\) \(0.567102 + 1.19603i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 0.697i)T \)
good7 \( 1 + 4.25T + 7T^{2} \)
11 \( 1 + (-1.51 - 4.64i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.43 - 4.41i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.815 + 0.592i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1 - 0.726i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.38 - 4.25i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.42 + 2.48i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.826 + 0.600i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.31 + 10.1i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.42 - 4.37i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (1.63 - 1.19i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.94 + 6.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.656 + 2.02i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.19 - 3.68i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.03 + 2.20i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-10.1 + 7.37i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.05 - 3.26i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.16 - 3.75i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.70 - 4.14i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.693 + 2.13i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.49 + 4.71i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.57087425429024193458211465624, −9.979351688805948358518141268993, −9.680608839675502827492444883658, −8.979990870325506708196699496180, −7.25968213230669181102505350128, −6.77554334548274575146013996538, −6.02240997842287599651467215896, −4.80120185809789808912304762092, −3.57904014708022523152501215852, −2.16990555146744312137114656798, 0.77287562021021133381345394851, 2.73694082955226081580857377180, 3.42582957246414247373655353715, 5.04306839518127450654043331509, 6.01652853348821086110203990849, 6.66763978006001052514672877319, 8.436295100081029084785539353579, 9.120936831451739957800753948076, 10.11326776245233400630265572546, 10.46365720231189566586893393899

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