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Properties

Label 2-450-25.4-c1-0-8
Degree $2$
Conductor $450$
Sign $-0.126 + 0.991i$
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 + (0.420 − 2.19i)5-s + 0.407i·7-s + (−0.587 − 0.809i)8-s + (−1.07 + 1.95i)10-s + (0.930 − 2.86i)11-s + (−0.172 + 0.0560i)13-s + (0.125 − 0.387i)14-s + (0.309 + 0.951i)16-s + (−1.82 − 2.50i)17-s + (1.15 − 0.840i)19-s + (1.63 − 1.52i)20-s + (−1.76 + 2.43i)22-s + (−1.06 − 0.345i)23-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.404 + 0.293i)4-s + (0.187 − 0.982i)5-s + 0.153i·7-s + (−0.207 − 0.286i)8-s + (−0.340 + 0.619i)10-s + (0.280 − 0.863i)11-s + (−0.0478 + 0.0155i)13-s + (0.0336 − 0.103i)14-s + (0.0772 + 0.237i)16-s + (−0.441 − 0.607i)17-s + (0.265 − 0.192i)19-s + (0.364 − 0.342i)20-s + (−0.377 + 0.519i)22-s + (−0.221 − 0.0719i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.126 + 0.991i$
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: \((2,\ 450,\ (\ :1/2),\ -0.126 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624550 - 0.709130i\)
\(L(\frac12)\) \(\approx\) \(0.624550 - 0.709130i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.420 + 2.19i)T \)
good7 \( 1 - 0.407iT - 7T^{2} \)
11 \( 1 + (-0.930 + 2.86i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.172 - 0.0560i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.82 + 2.50i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.15 + 0.840i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.06 + 0.345i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.127 - 0.0928i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-6.96 + 5.06i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.19 - 1.36i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.54 + 7.83i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (-1.62 + 2.23i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.93 - 6.78i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.21 - 6.82i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.09 + 6.44i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.32 - 3.19i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (2.49 + 1.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.18 + 2.66i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.59 - 4.06i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.70 - 11.9i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.52 - 13.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (9.59 - 13.2i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.78148844073038458842630298776, −9.816625606960212277676328756326, −8.943192664137215669604203097942, −8.459253255110493085184107488555, −7.34180091600589257692955304074, −6.16571771431642530152800512630, −5.14264488849334947332000450185, −3.84553432648377965224055408966, −2.32966576420884372838173850544, −0.75120562852821136164952261690, 1.77963080125408639924540076625, 3.13144509715714392352686431111, 4.58613228793491371218418028416, 6.05660772968135038180749508965, 6.78934073191255347728176040536, 7.60924172610459009313463236175, 8.591744716706328896894961294153, 9.767350433997565085230907105767, 10.21539885004215739009445199115, 11.15416166147401663933324259892

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