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Properties

Label 2-450-225.106-c1-0-9
Degree $2$
Conductor $450$
Sign $0.276 - 0.960i$
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  + (−0.978 − 0.207i)2-s + (0.271 + 1.71i)3-s + (0.913 + 0.406i)4-s + (2.23 − 0.0445i)5-s + (0.0900 − 1.72i)6-s + (−0.735 + 1.27i)7-s + (−0.809 − 0.587i)8-s + (−2.85 + 0.928i)9-s + (−2.19 − 0.421i)10-s + (4.34 + 0.924i)11-s + (−0.447 + 1.67i)12-s + (2.63 − 0.560i)13-s + (0.984 − 1.09i)14-s + (0.683 + 3.81i)15-s + (0.669 + 0.743i)16-s + (3.19 + 2.32i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.147i)2-s + (0.156 + 0.987i)3-s + (0.456 + 0.203i)4-s + (0.999 − 0.0199i)5-s + (0.0367 − 0.706i)6-s + (−0.278 + 0.481i)7-s + (−0.286 − 0.207i)8-s + (−0.950 + 0.309i)9-s + (−0.694 − 0.133i)10-s + (1.31 + 0.278i)11-s + (−0.129 + 0.483i)12-s + (0.730 − 0.155i)13-s + (0.263 − 0.292i)14-s + (0.176 + 0.984i)15-s + (0.167 + 0.185i)16-s + (0.775 + 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.276 - 0.960i$
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: \((2,\ 450,\ (\ :1/2),\ 0.276 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00898 + 0.759266i\)
\(L(\frac12)\) \(\approx\) \(1.00898 + 0.759266i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.271 - 1.71i)T \)
5 \( 1 + (-2.23 + 0.0445i)T \)
good7 \( 1 + (0.735 - 1.27i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.34 - 0.924i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-2.63 + 0.560i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-3.19 - 2.32i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.20 + 3.78i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (5.28 - 5.87i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (-0.418 - 3.98i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-1.01 + 9.70i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (1.02 - 3.15i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.79 + 0.807i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (5.75 - 9.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.795 - 7.56i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-8.00 + 5.81i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.09 - 0.870i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (7.94 + 1.68i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-0.460 + 4.37i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (-5.21 + 3.78i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.30 + 13.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.270 + 2.57i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-2.12 + 0.945i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (3.16 + 9.74i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.810 + 7.70i)T + (-94.8 + 20.1i)T^{2} \)
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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

−11.03180810779547172718524054639, −10.16028721055238369864379031025, −9.409335874305772863711297741262, −9.000248467195203688577649616123, −7.965069835971145265862666283586, −6.33858782603543688129260686959, −5.87048881548878736861789333459, −4.35096611059656685828000581176, −3.15919264838824285126654919499, −1.77237968660561077521427514132, 1.07554890626148051035916134326, 2.20662549388390185863590103443, 3.75366398193373463888480346559, 5.73270225039409665339069809384, 6.44901141645431425706086516429, 7.03532695200728644325020673634, 8.414419581005899510872980916412, 8.845586037980496022243005813592, 10.00403877425969566019598884780, 10.67523663972014519628487403453

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