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Properties

Label 2-450-45.32-c1-0-1
Degree $2$
Conductor $450$
Sign $-0.0664 - 0.997i$
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  + (−0.965 − 0.258i)2-s + (1.33 + 1.10i)3-s + (0.866 + 0.499i)4-s + (−1 − 1.41i)6-s + (−0.283 + 1.05i)7-s + (−0.707 − 0.707i)8-s + (0.548 + 2.94i)9-s + (−5.44 + 3.14i)11-s + (0.599 + 1.62i)12-s + (0.896 + 3.34i)13-s + (0.548 − 0.949i)14-s + (0.500 + 0.866i)16-s + (3.14 − 3.14i)17-s + (0.233 − 2.99i)18-s + 1.55i·19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.769 + 0.639i)3-s + (0.433 + 0.249i)4-s + (−0.408 − 0.577i)6-s + (−0.107 + 0.400i)7-s + (−0.249 − 0.249i)8-s + (0.182 + 0.983i)9-s + (−1.64 + 0.948i)11-s + (0.173 + 0.469i)12-s + (0.248 + 0.928i)13-s + (0.146 − 0.253i)14-s + (0.125 + 0.216i)16-s + (0.763 − 0.763i)17-s + (0.0551 − 0.704i)18-s + 0.355i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0664 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0664 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0664 - 0.997i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.0664 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759228 + 0.811490i\)
\(L(\frac12)\) \(\approx\) \(0.759228 + 0.811490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.33 - 1.10i)T \)
5 \( 1 \)
good7 \( 1 + (0.283 - 1.05i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.44 - 3.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.896 - 3.34i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3.14 + 3.14i)T - 17iT^{2} \)
19 \( 1 - 1.55iT - 19T^{2} \)
23 \( 1 + (0.965 - 0.258i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.57 - 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.22 + 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (3.39 + 1.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.34 - 0.896i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (8.69 + 2.32i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.61 - 6.61i)T + 53iT^{2} \)
59 \( 1 + (-5.90 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.72 - 4.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.65 + 0.978i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 + (2.89 - 2.89i)T - 73iT^{2} \)
79 \( 1 + (2.12 - 1.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.142 + 0.531i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.36T + 89T^{2} \)
97 \( 1 + (-2.89 + 10.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.07898502706683029125725659680, −10.04782717927924689713456422306, −9.731473077506928056880971612586, −8.671931408375978309704487102706, −7.893168590862399811556554605859, −7.09371856207347281397334758829, −5.51198100109358302638751335867, −4.44952686305174845534160114951, −3.03043074605402081035950539876, −2.08814307264416714205538695366, 0.78498707918688162994646362320, 2.52668089245270803036630115122, 3.50612487407875756643298282684, 5.40180441342406082084250872479, 6.35868876425416823285656399900, 7.55703022481491181026247470830, 8.093077146260975022401179856684, 8.726648252456857748864249732865, 10.07419653926218956342135259164, 10.50409851319560368478885006638

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