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Properties

Label 2-450-25.4-c1-0-11
Degree $2$
Conductor $450$
Sign $0.951 + 0.308i$
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 + (1.95 − 1.07i)5-s − 3.03i·7-s + (0.587 + 0.809i)8-s + (2.19 − 0.420i)10-s + (0.791 − 2.43i)11-s + (−0.945 + 0.307i)13-s + (0.938 − 2.88i)14-s + (0.309 + 0.951i)16-s + (−0.558 − 0.769i)17-s + (−5.39 + 3.91i)19-s + (2.21 + 0.279i)20-s + (1.50 − 2.07i)22-s + (5.81 + 1.89i)23-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.404 + 0.293i)4-s + (0.876 − 0.482i)5-s − 1.14i·7-s + (0.207 + 0.286i)8-s + (0.694 − 0.132i)10-s + (0.238 − 0.734i)11-s + (−0.262 + 0.0852i)13-s + (0.250 − 0.771i)14-s + (0.0772 + 0.237i)16-s + (−0.135 − 0.186i)17-s + (−1.23 + 0.898i)19-s + (0.496 + 0.0624i)20-s + (0.320 − 0.441i)22-s + (1.21 + 0.394i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.951 + 0.308i$
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: \((2,\ 450,\ (\ :1/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29097 - 0.361677i\)
\(L(\frac12)\) \(\approx\) \(2.29097 - 0.361677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 + 1.07i)T \)
good7 \( 1 + 3.03iT - 7T^{2} \)
11 \( 1 + (-0.791 + 2.43i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.945 - 0.307i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.558 + 0.769i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.39 - 3.91i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.81 - 1.89i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-7.82 - 5.68i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.65 - 4.10i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.94 - 0.955i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.167 - 0.516i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.08iT - 43T^{2} \)
47 \( 1 + (1.62 - 2.23i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.08 + 1.49i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.998 + 3.07i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.68 - 11.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.32 + 10.0i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (8.64 + 6.28i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-13.7 - 4.47i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.3 + 8.95i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.22 - 7.19i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.76 - 8.50i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.835 - 1.15i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.86297618022623256326121668475, −10.42295885473816118837106960825, −9.179810708549254155559513828681, −8.318201513180169774452950096436, −7.08050598922262553869432115183, −6.34505741363401629420166109042, −5.25569622145791460317203429230, −4.33943176142237433600231189734, −3.12277188851830906478437698337, −1.42233520161699651508001104645, 2.07079002233216617536877956899, 2.79014248027472242235630775604, 4.44009640705517156527091956150, 5.41841626202520400851005657045, 6.33954601752315970911488000319, 7.08041293118014670592009435358, 8.636143986642492422719843729811, 9.417081626754555472141841205262, 10.36376887992869689799294517802, 11.16636668003304723900289495849

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