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Properties

Label 2-450-225.79-c1-0-4
Degree $2$
Conductor $450$
Sign $-0.866 - 0.498i$
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  + (−0.743 + 0.669i)2-s + (0.547 + 1.64i)3-s + (0.104 − 0.994i)4-s + (−0.449 + 2.19i)5-s + (−1.50 − 0.854i)6-s + (2.70 + 1.56i)7-s + (0.587 + 0.809i)8-s + (−2.40 + 1.79i)9-s + (−1.13 − 1.92i)10-s + (−1.59 − 1.77i)11-s + (1.69 − 0.372i)12-s + (3.97 + 3.57i)13-s + (−3.05 + 0.648i)14-s + (−3.84 + 0.461i)15-s + (−0.978 − 0.207i)16-s + (−1.15 − 1.59i)17-s + ⋯
L(s)  = 1  + (−0.525 + 0.473i)2-s + (0.316 + 0.948i)3-s + (0.0522 − 0.497i)4-s + (−0.200 + 0.979i)5-s + (−0.615 − 0.348i)6-s + (1.02 + 0.589i)7-s + (0.207 + 0.286i)8-s + (−0.800 + 0.599i)9-s + (−0.357 − 0.609i)10-s + (−0.482 − 0.535i)11-s + (0.488 − 0.107i)12-s + (1.10 + 0.991i)13-s + (−0.815 + 0.173i)14-s + (−0.992 + 0.119i)15-s + (−0.244 − 0.0519i)16-s + (−0.280 − 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.866 - 0.498i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.866 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.302539 + 1.13273i\)
\(L(\frac12)\) \(\approx\) \(0.302539 + 1.13273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.743 - 0.669i)T \)
3 \( 1 + (-0.547 - 1.64i)T \)
5 \( 1 + (0.449 - 2.19i)T \)
good7 \( 1 + (-2.70 - 1.56i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.59 + 1.77i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-3.97 - 3.57i)T + (1.35 + 12.9i)T^{2} \)
17 \( 1 + (1.15 + 1.59i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.947 + 0.688i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.199 + 0.937i)T + (-21.0 + 9.35i)T^{2} \)
29 \( 1 + (4.42 - 1.97i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (-6.48 - 2.88i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (2.31 - 0.752i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-4.00 + 4.45i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (10.7 + 6.20i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.99 - 6.72i)T + (-31.4 + 34.9i)T^{2} \)
53 \( 1 + (-4.28 + 5.89i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (6.20 - 6.89i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (2.34 + 2.60i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (-2.52 + 5.66i)T + (-44.8 - 49.7i)T^{2} \)
71 \( 1 + (-10.3 - 7.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-11.8 - 3.86i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.81 - 3.03i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-16.4 + 1.72i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + (4.58 - 14.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.465 - 1.04i)T + (-64.9 + 72.0i)T^{2} \)
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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

−11.11707575963561874101052667753, −10.67294807754902114593190819462, −9.561635291940205131189316813388, −8.642267763666714268014465734107, −8.127956674714697734841143179168, −6.91796988596416810201832928223, −5.82574664623695585381904987368, −4.82811018351994880433155944184, −3.55206732377104980157373764318, −2.20608713387128381323219947795, 0.889621416283906944873611011931, 1.92186329058226776040480450267, 3.55674701809446977433281673425, 4.82081040919482224803082466662, 6.08120758307029962962318613110, 7.55134547445796250688295259347, 8.036970377444920925217272959791, 8.623072078942828516431733547529, 9.758712132048469175397068470338, 10.87093533738640342867661750978

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