Properties

Label 2-450-225.106-c1-0-8
Degree $2$
Conductor $450$
Sign $0.904 - 0.425i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.978 + 0.207i)2-s + (−1.68 − 0.411i)3-s + (0.913 + 0.406i)4-s + (−2.23 − 0.0183i)5-s + (−1.56 − 0.752i)6-s + (−0.0474 + 0.0821i)7-s + (0.809 + 0.587i)8-s + (2.66 + 1.38i)9-s + (−2.18 − 0.482i)10-s + (4.91 + 1.04i)11-s + (−1.36 − 1.06i)12-s + (2.77 − 0.590i)13-s + (−0.0635 + 0.0705i)14-s + (3.75 + 0.950i)15-s + (0.669 + 0.743i)16-s + (1.71 + 1.24i)17-s + ⋯
L(s)  = 1  + (0.691 + 0.147i)2-s + (−0.971 − 0.237i)3-s + (0.456 + 0.203i)4-s + (−0.999 − 0.00821i)5-s + (−0.636 − 0.307i)6-s + (−0.0179 + 0.0310i)7-s + (0.286 + 0.207i)8-s + (0.887 + 0.461i)9-s + (−0.690 − 0.152i)10-s + (1.48 + 0.314i)11-s + (−0.395 − 0.306i)12-s + (0.771 − 0.163i)13-s + (−0.0169 + 0.0188i)14-s + (0.969 + 0.245i)15-s + (0.167 + 0.185i)16-s + (0.416 + 0.302i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.904 - 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43962 + 0.321607i\)
\(L(\frac12)\) \(\approx\) \(1.43962 + 0.321607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (1.68 + 0.411i)T \)
5 \( 1 + (2.23 + 0.0183i)T \)
good7 \( 1 + (0.0474 - 0.0821i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.91 - 1.04i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-2.77 + 0.590i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-1.71 - 1.24i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.22 - 2.34i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.88 - 3.19i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (-0.776 - 7.39i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-1.14 + 10.8i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-1.52 + 4.68i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.31 - 0.705i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (3.21 - 5.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.824 + 7.84i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (0.715 - 0.520i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (9.73 - 2.06i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-10.6 - 2.26i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (0.504 - 4.79i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (-1.84 + 1.34i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.793 - 2.44i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.30 + 12.4i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-9.05 + 4.03i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-2.31 - 7.13i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.804 + 7.65i)T + (-94.8 + 20.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.55632784251831172475355132679, −10.62307327239519251887944337884, −9.446414163838645038206213817993, −8.088855743059706876913123109336, −7.27113179153355031685727981842, −6.37948798805499335055225185462, −5.52443185459555372516280643027, −4.27091235118660342905434139411, −3.59447826937218038088243298374, −1.36329252336223837735797134269, 1.07090770302262265086422751794, 3.40820332390711155313236890025, 4.17729370022734715603200245470, 5.13443530510175964229612304045, 6.38681408816962518159887687189, 6.89472670568897532729598047981, 8.226181158164245806439698096691, 9.383976307755826949592547007418, 10.48206796727020895021648489587, 11.33576404723897773916574924750

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