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Properties

Label 2-450-225.83-c1-0-24
Degree $2$
Conductor $450$
Sign $-0.309 + 0.950i$
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  + (−0.544 − 0.838i)2-s + (1.52 − 0.828i)3-s + (−0.406 + 0.913i)4-s + (1.49 − 1.66i)5-s + (−1.52 − 0.823i)6-s + (−0.322 − 1.20i)7-s + (0.987 − 0.156i)8-s + (1.62 − 2.52i)9-s + (−2.20 − 0.346i)10-s + (−0.109 + 0.517i)11-s + (0.138 + 1.72i)12-s + (−3.10 − 2.01i)13-s + (−0.833 + 0.925i)14-s + (0.891 − 3.76i)15-s + (−0.669 − 0.743i)16-s + (0.976 + 6.16i)17-s + ⋯
L(s)  = 1  + (−0.385 − 0.593i)2-s + (0.878 − 0.478i)3-s + (−0.203 + 0.456i)4-s + (0.667 − 0.744i)5-s + (−0.621 − 0.336i)6-s + (−0.121 − 0.454i)7-s + (0.349 − 0.0553i)8-s + (0.541 − 0.840i)9-s + (−0.698 − 0.109i)10-s + (−0.0331 + 0.155i)11-s + (0.0400 + 0.498i)12-s + (−0.861 − 0.559i)13-s + (−0.222 + 0.247i)14-s + (0.230 − 0.973i)15-s + (−0.167 − 0.185i)16-s + (0.236 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.309 + 0.950i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946936 - 1.30394i\)
\(L(\frac12)\) \(\approx\) \(0.946936 - 1.30394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.544 + 0.838i)T \)
3 \( 1 + (-1.52 + 0.828i)T \)
5 \( 1 + (-1.49 + 1.66i)T \)
good7 \( 1 + (0.322 + 1.20i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.109 - 0.517i)T + (-10.0 - 4.47i)T^{2} \)
13 \( 1 + (3.10 + 2.01i)T + (5.28 + 11.8i)T^{2} \)
17 \( 1 + (-0.976 - 6.16i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-2.84 + 3.91i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.265 - 5.05i)T + (-22.8 + 2.40i)T^{2} \)
29 \( 1 + (0.817 + 7.77i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (0.496 - 4.72i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-2.15 - 4.22i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-0.0520 - 0.244i)T + (-37.4 + 16.6i)T^{2} \)
43 \( 1 + (-2.07 + 0.555i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.92 - 6.07i)T + (-9.77 - 45.9i)T^{2} \)
53 \( 1 + (0.615 - 3.88i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-3.79 + 0.807i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (13.3 + 2.82i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (4.51 + 5.57i)T + (-13.9 + 65.5i)T^{2} \)
71 \( 1 + (-5.74 - 7.90i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.19 + 2.34i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-14.9 + 1.56i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (0.773 - 2.01i)T + (-61.6 - 55.5i)T^{2} \)
89 \( 1 + (-4.13 - 12.7i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.576 + 0.466i)T + (20.1 + 94.8i)T^{2} \)
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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

−10.55936168252634051759482228911, −9.735825148441142910273817510445, −9.210814966262003368332490525424, −8.133706974084376473306596887888, −7.52115215157336383721998300722, −6.23360573686126083695536878075, −4.84161061739790394323701022355, −3.58863967530522700487298435656, −2.34804989373866294780027057570, −1.14138273668407761936040167712, 2.14396552281537865483024462680, 3.19176576518754955452569503813, 4.75998248257641039520207512064, 5.73577472257742887790928852369, 6.99516231294687816474788631807, 7.60842773285584006952626053799, 8.847962305450618715365027354137, 9.497542116896537654958063667874, 10.08276416563118651776165359271, 11.03896845099089344819591314390

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