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Properties

Label 2-450-45.4-c1-0-6
Degree $2$
Conductor $450$
Sign $0.972 - 0.232i$
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  + (0.866 − 0.5i)2-s + (0.866 + 1.5i)3-s + (0.499 − 0.866i)4-s + (1.5 + 0.866i)6-s + (1.73 − i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (1.5 + 2.59i)11-s + 1.73·12-s + (1.73 + i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s − 3i·17-s + 3i·18-s + 19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 − 0.433i)4-s + (0.612 + 0.353i)6-s + (0.654 − 0.377i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.452 + 0.783i)11-s + 0.499·12-s + (0.480 + 0.277i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s − 0.727i·17-s + 0.707i·18-s + 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.972 - 0.232i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.972 - 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40303 + 0.283034i\)
\(L(\frac12)\) \(\approx\) \(2.40303 + 0.283034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
good7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 - 6i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-4.33 + 2.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.18097805283016277579210671583, −10.23031766895909532453799949936, −9.565730512945298119363828487064, −8.517911713700305473509675968795, −7.51375626212334676727221757919, −6.28542156010689608043724704067, −4.91058330644805171951861617830, −4.36981676749486436795081536550, −3.24948465149778468425840071771, −1.86232344382324101445466843707, 1.58053199738587509054199677009, 3.00538038939824399320404145252, 4.10147375122452446700351984605, 5.64503000377898697053221655777, 6.25029691053776191199681850829, 7.38332947979429658778190626789, 8.324903060608662562686265802093, 8.733703047672897402133758167910, 10.21328161457640303531678489000, 11.54634357293725921538241736585

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