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Properties

Label 2-450-225.38-c1-0-23
Degree $2$
Conductor $450$
Sign $-0.952 - 0.304i$
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  + (−0.933 − 0.358i)2-s + (−1.70 − 0.286i)3-s + (0.743 + 0.669i)4-s + (−0.417 − 2.19i)5-s + (1.49 + 0.879i)6-s + (−0.520 − 1.94i)7-s + (−0.453 − 0.891i)8-s + (2.83 + 0.979i)9-s + (−0.397 + 2.20i)10-s + (−0.844 − 1.89i)11-s + (−1.07 − 1.35i)12-s + (0.114 + 0.297i)13-s + (−0.210 + 1.99i)14-s + (0.0833 + 3.87i)15-s + (0.104 + 0.994i)16-s + (−1.68 + 0.857i)17-s + ⋯
L(s)  = 1  + (−0.660 − 0.253i)2-s + (−0.986 − 0.165i)3-s + (0.371 + 0.334i)4-s + (−0.186 − 0.982i)5-s + (0.609 + 0.359i)6-s + (−0.196 − 0.733i)7-s + (−0.160 − 0.315i)8-s + (0.945 + 0.326i)9-s + (−0.125 + 0.695i)10-s + (−0.254 − 0.571i)11-s + (−0.311 − 0.391i)12-s + (0.0317 + 0.0825i)13-s + (−0.0561 + 0.534i)14-s + (0.0215 + 0.999i)15-s + (0.0261 + 0.248i)16-s + (−0.408 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.952 - 0.304i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0428404 + 0.274599i\)
\(L(\frac12)\) \(\approx\) \(0.0428404 + 0.274599i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.933 + 0.358i)T \)
3 \( 1 + (1.70 + 0.286i)T \)
5 \( 1 + (0.417 + 2.19i)T \)
good7 \( 1 + (0.520 + 1.94i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.844 + 1.89i)T + (-7.36 + 8.17i)T^{2} \)
13 \( 1 + (-0.114 - 0.297i)T + (-9.66 + 8.69i)T^{2} \)
17 \( 1 + (1.68 - 0.857i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-3.79 - 1.23i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.87 + 4.78i)T + (-4.78 + 22.4i)T^{2} \)
29 \( 1 + (3.27 + 0.695i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (7.45 - 1.58i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (1.48 - 9.36i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.41 - 5.43i)T + (-27.4 - 30.4i)T^{2} \)
43 \( 1 + (3.97 - 1.06i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (6.72 + 10.3i)T + (-19.1 + 42.9i)T^{2} \)
53 \( 1 + (-7.22 - 3.68i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-2.30 - 1.02i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-0.462 + 0.205i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (4.30 - 6.62i)T + (-27.2 - 61.2i)T^{2} \)
71 \( 1 + (8.25 - 2.68i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.01 + 12.7i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.97 + 9.30i)T + (-72.1 - 32.1i)T^{2} \)
83 \( 1 + (-8.75 - 0.458i)T + (82.5 + 8.67i)T^{2} \)
89 \( 1 + (-1.08 - 0.785i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.33 + 2.81i)T + (39.4 - 88.6i)T^{2} \)
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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

−10.51126697188928324316743609664, −9.933862607743608640629898701950, −8.784699103280194201277322855209, −7.900508743145034922494327748612, −6.97405805422877856827491974678, −5.90331942745183404654065595903, −4.82349545165270262376563621723, −3.68586130212310093730321744372, −1.57102986472501202210607078665, −0.24279347571114975553841010169, 2.09459382713844063594627655619, 3.68779293776003392103213610804, 5.28965742104555268182131503217, 6.00176845181409584701860299785, 7.09316162900011622276448243800, 7.60276095419721853249355414812, 9.155943944539791146607379628310, 9.812419763571949954929027989820, 10.70501823768714480879107110791, 11.41020983280246068550502287043

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