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Properties

Label 2-450-225.2-c1-0-28
Degree $2$
Conductor $450$
Sign $-0.673 - 0.739i$
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  + (−0.777 − 0.629i)2-s + (−0.942 − 1.45i)3-s + (0.207 + 0.978i)4-s + (1.99 + 1.01i)5-s + (−0.181 + 1.72i)6-s + (−1.72 + 0.461i)7-s + (0.453 − 0.891i)8-s + (−1.22 + 2.73i)9-s + (−0.912 − 2.04i)10-s + (−4.41 + 0.464i)11-s + (1.22 − 1.22i)12-s + (−1.01 − 1.25i)13-s + (1.62 + 0.724i)14-s + (−0.409 − 3.85i)15-s + (−0.913 + 0.406i)16-s + (−5.47 − 2.78i)17-s + ⋯
L(s)  = 1  + (−0.549 − 0.444i)2-s + (−0.544 − 0.838i)3-s + (0.103 + 0.489i)4-s + (0.891 + 0.452i)5-s + (−0.0742 + 0.703i)6-s + (−0.650 + 0.174i)7-s + (0.160 − 0.315i)8-s + (−0.407 + 0.913i)9-s + (−0.288 − 0.645i)10-s + (−1.33 + 0.139i)11-s + (0.353 − 0.353i)12-s + (−0.281 − 0.347i)13-s + (0.434 + 0.193i)14-s + (−0.105 − 0.994i)15-s + (−0.228 + 0.101i)16-s + (−1.32 − 0.676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.673 - 0.739i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00307292 + 0.00695693i\)
\(L(\frac12)\) \(\approx\) \(0.00307292 + 0.00695693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.777 + 0.629i)T \)
3 \( 1 + (0.942 + 1.45i)T \)
5 \( 1 + (-1.99 - 1.01i)T \)
good7 \( 1 + (1.72 - 0.461i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.41 - 0.464i)T + (10.7 - 2.28i)T^{2} \)
13 \( 1 + (1.01 + 1.25i)T + (-2.70 + 12.7i)T^{2} \)
17 \( 1 + (5.47 + 2.78i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (3.58 - 1.16i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.64 + 4.29i)T + (-17.0 + 15.3i)T^{2} \)
29 \( 1 + (3.83 + 4.25i)T + (-3.03 + 28.8i)T^{2} \)
31 \( 1 + (4.81 - 5.35i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-1.49 - 9.43i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-10.2 - 1.07i)T + (40.1 + 8.52i)T^{2} \)
43 \( 1 + (1.25 + 4.67i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.88 + 0.255i)T + (46.7 + 4.91i)T^{2} \)
53 \( 1 + (0.446 - 0.227i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-0.344 + 3.28i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (-1.05 - 10.0i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-11.1 + 0.582i)T + (66.6 - 7.00i)T^{2} \)
71 \( 1 + (3.66 + 1.19i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.95 + 12.3i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-0.600 + 0.540i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (4.97 + 7.65i)T + (-33.7 + 75.8i)T^{2} \)
89 \( 1 + (-7.69 + 5.58i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.188 + 3.59i)T + (-96.4 - 10.1i)T^{2} \)
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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

−10.57608141481895263530443045298, −9.845099116625675892255690193289, −8.755925909062050496440217776192, −7.72283650601046584533548375511, −6.78026310794314572185521613897, −6.03269943984225385060672189486, −4.87626341660696770425089302475, −2.79713655105799944782770305616, −2.10660346370516398362017725667, −0.00541780292562631680487768290, 2.28211955622855599915546630341, 4.08387636696766475732666537439, 5.25716996540143332036794095523, 5.95168281329145360573576086760, 6.86717397578232355289275666969, 8.215093643564725402556724142802, 9.311052722042557008287144063518, 9.614020603007759801395930586493, 10.73805435810602417912370339669, 11.08614535551434431750397705178

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