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Properties

Label 2-450-25.6-c1-0-4
Degree $2$
Conductor $450$
Sign $-0.367 - 0.929i$
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 + (−1.01 + 1.99i)5-s + 4.77·7-s + (−0.809 − 0.587i)8-s + (−2.20 − 0.350i)10-s + (−0.0788 − 0.242i)11-s + (−1.32 + 4.07i)13-s + (1.47 + 4.54i)14-s + (0.309 − 0.951i)16-s + (−1.88 − 1.37i)17-s + (1.12 + 0.814i)19-s + (−0.349 − 2.20i)20-s + (0.206 − 0.149i)22-s + (2.31 + 7.11i)23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.454 + 0.890i)5-s + 1.80·7-s + (−0.286 − 0.207i)8-s + (−0.698 − 0.110i)10-s + (−0.0237 − 0.0731i)11-s + (−0.367 + 1.13i)13-s + (0.394 + 1.21i)14-s + (0.0772 − 0.237i)16-s + (−0.457 − 0.332i)17-s + (0.257 + 0.186i)19-s + (−0.0781 − 0.493i)20-s + (0.0440 − 0.0319i)22-s + (0.481 + 1.48i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.367 - 0.929i$
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: \((2,\ 450,\ (\ :1/2),\ -0.367 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.867944 + 1.27677i\)
\(L(\frac12)\) \(\approx\) \(0.867944 + 1.27677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 - 1.99i)T \)
good7 \( 1 - 4.77T + 7T^{2} \)
11 \( 1 + (0.0788 + 0.242i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.32 - 4.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.88 + 1.37i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.12 - 0.814i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.31 - 7.11i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.13 - 3.73i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.143 + 0.103i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.57 + 4.84i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.681 - 2.09i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.60T + 43T^{2} \)
47 \( 1 + (-6.09 + 4.42i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-10.9 + 7.93i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.20 + 6.78i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.67 - 11.3i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.44 + 4.68i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-8.27 + 6.01i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.98 + 9.18i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.91 + 2.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.8 + 9.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-3.45 - 10.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-12.7 + 9.26i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.51171026218821440382117694884, −10.69626110320276745497375433728, −9.373085745770757655243832098506, −8.446322258967035689469179911624, −7.43961922632334980073860816634, −7.07571368414115973392464223813, −5.62821516436153132050464739264, −4.70613670232731574270359050658, −3.69186821589191175147643954623, −1.99179712414074287173455813852, 0.996263077541253907202289458723, 2.39556505843421011222563495015, 4.10854413487987975493209961713, 4.84520632278216943881578406981, 5.61146859294789863958949503201, 7.42936943074736011303813769219, 8.281855390435629114211738767334, 8.831830275972703677854691576069, 10.14109665546659869398420070991, 11.01445741192111882553718735079

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