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Properties

Label 2-450-9.7-c1-0-0
Degree $2$
Conductor $450$
Sign $-0.00922 - 0.999i$
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  + (0.5 − 0.866i)2-s + (−1 − 1.41i)3-s + (−0.499 − 0.866i)4-s + (−1.72 + 0.158i)6-s + (−2.22 + 3.85i)7-s − 0.999·8-s + (−1.00 + 2.82i)9-s + (−2.44 + 4.24i)11-s + (−0.724 + 1.57i)12-s + (−2.22 − 3.85i)13-s + (2.22 + 3.85i)14-s + (−0.5 + 0.866i)16-s − 4.89·17-s + (1.94 + 2.28i)18-s + 2.55·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.577 − 0.816i)3-s + (−0.249 − 0.433i)4-s + (−0.704 + 0.0648i)6-s + (−0.840 + 1.45i)7-s − 0.353·8-s + (−0.333 + 0.942i)9-s + (−0.738 + 1.27i)11-s + (−0.209 + 0.454i)12-s + (−0.617 − 1.06i)13-s + (0.594 + 1.02i)14-s + (−0.125 + 0.216i)16-s − 1.18·17-s + (0.459 + 0.537i)18-s + 0.585·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00922 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00922 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.00922 - 0.999i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.00922 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195448 + 0.197261i\)
\(L(\frac12)\) \(\approx\) \(0.195448 + 0.197261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (1 + 1.41i)T \)
5 \( 1 \)
good7 \( 1 + (2.22 - 3.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.44 - 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.22 + 3.85i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (1.22 + 2.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.224 - 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.72 - 6.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.550 + 0.953i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.44T + 53T^{2} \)
59 \( 1 + (-0.275 - 0.476i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.17 - 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.34T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (-6.34 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.275 + 0.476i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-5.39 + 9.35i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.59935443102799556720897023348, −10.49783995419720147800210628516, −9.741820776546396445898675744147, −8.684451489671350105430276794906, −7.53221918263658036481111378447, −6.49775552777731365659975152227, −5.53648767503993290039723138987, −4.82668155520275815234302295139, −2.85701557261642448488863660861, −2.13103791530114760922457750126, 0.15362727295134014143632646292, 3.22649891428358958167800965498, 4.12172909379042164114756744856, 5.03327263863342651316763184421, 6.24429939898981645973790688711, 6.85609045178531231853459727632, 7.994536546975601016031700817780, 9.247977314449539292739841512016, 9.918336845885588198230216468723, 10.93123934084861435816468097557

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