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Properties

Label 2-450-45.23-c1-0-9
Degree $2$
Conductor $450$
Sign $0.976 + 0.216i$
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  + (0.258 − 0.965i)2-s + (1.22 + 1.22i)3-s + (−0.866 − 0.499i)4-s + (1.49 − 0.866i)6-s + (−0.707 + 0.707i)8-s + 2.99i·9-s + (3 − 1.73i)11-s + (−0.448 − 1.67i)12-s + (3.34 − 0.896i)13-s + (0.500 + 0.866i)16-s + (4.24 + 4.24i)17-s + (2.89 + 0.776i)18-s − 5i·19-s + (−0.896 − 3.34i)22-s + (1.55 + 5.79i)23-s − 1.73·24-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.707 + 0.707i)3-s + (−0.433 − 0.249i)4-s + (0.612 − 0.353i)6-s + (−0.249 + 0.249i)8-s + 0.999i·9-s + (0.904 − 0.522i)11-s + (−0.129 − 0.482i)12-s + (0.928 − 0.248i)13-s + (0.125 + 0.216i)16-s + (1.02 + 1.02i)17-s + (0.683 + 0.183i)18-s − 1.14i·19-s + (−0.191 − 0.713i)22-s + (0.323 + 1.20i)23-s − 0.353·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.976 + 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94421 - 0.212684i\)
\(L(\frac12)\) \(\approx\) \(1.94421 - 0.212684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.34 + 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + (-1.55 - 5.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.46 + 6i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.13 + 11.7i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.55 - 5.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.24 - 4.24i)T - 53iT^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.24 + 8.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (8.57 + 8.57i)T + 73iT^{2} \)
79 \( 1 + (-12.1 + 7i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.69 + 2.32i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-5.01 - 1.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.00261839807556554299400846438, −10.22746189247306308310181445413, −9.259987659572009470743776151091, −8.686821382674912564534771686759, −7.66794352671573191769276095400, −6.14544675611564555639389249135, −5.08416625251635074300425443153, −3.78934847837878129783839469170, −3.27349648521876616180423188654, −1.60130966687135647370339238158, 1.44539016421134540990098997469, 3.18827248221554172796612211487, 4.19792779091613723184056998377, 5.67972830068207190448255140767, 6.63778777365368414072487405139, 7.37026821791066049271604998861, 8.291757667534044262806203372516, 9.097121217827025093098388538216, 9.875526753924580103170241462284, 11.32598239231973914294400380658

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