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Properties

Label 2-450-25.4-c1-0-6
Degree $2$
Conductor $450$
Sign $0.502 - 0.864i$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (2.06 + 0.847i)5-s + 4.07i·7-s + (0.587 + 0.809i)8-s + (1.70 + 1.44i)10-s + (1.01 − 3.12i)11-s + (−5.15 + 1.67i)13-s + (−1.26 + 3.87i)14-s + (0.309 + 0.951i)16-s + (−2.03 − 2.79i)17-s + (1.32 − 0.961i)19-s + (1.17 + 1.90i)20-s + (1.93 − 2.65i)22-s + (0.581 + 0.188i)23-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.404 + 0.293i)4-s + (0.925 + 0.379i)5-s + 1.54i·7-s + (0.207 + 0.286i)8-s + (0.539 + 0.457i)10-s + (0.305 − 0.941i)11-s + (−1.43 + 0.464i)13-s + (−0.336 + 1.03i)14-s + (0.0772 + 0.237i)16-s + (−0.493 − 0.678i)17-s + (0.303 − 0.220i)19-s + (0.262 + 0.425i)20-s + (0.411 − 0.566i)22-s + (0.121 + 0.0393i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98112 + 1.14000i\)
\(L(\frac12)\) \(\approx\) \(1.98112 + 1.14000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 \)
5 \( 1 + (-2.06 - 0.847i)T \)
good7 \( 1 - 4.07iT - 7T^{2} \)
11 \( 1 + (-1.01 + 3.12i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (5.15 - 1.67i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.03 + 2.79i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.32 + 0.961i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.581 - 0.188i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.78 - 2.74i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-6.71 + 4.88i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.28 + 0.741i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.905 + 2.78i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.64iT - 43T^{2} \)
47 \( 1 + (-4.06 + 5.59i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.99 - 5.49i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.38 + 13.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.88 - 11.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.39 - 8.79i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (7.33 + 5.32i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.65 + 2.81i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.31 + 2.40i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.19 + 5.76i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.48 + 4.56i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-8.67 + 11.9i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.61827652534438107441506224182, −10.35375568902434299527704313853, −9.309990245944769476071558273038, −8.716877568485857990667864863428, −7.29258324036618529334090441029, −6.34172652543735360533609788389, −5.58983608565151432872136184451, −4.75160734853668371098797606249, −2.94903789574831393194510162154, −2.26438542943697736968498208351, 1.34205802167370427709055950119, 2.76674852979827351002450562637, 4.36625168588765561632808297057, 4.84791362889025501256095932220, 6.25228306608696419758281422423, 7.05286783468978259626007941740, 8.021364360654748935173210339045, 9.605031809063287317026001786503, 10.06469885773240803208807901825, 10.78974250154803586233368843384

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