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Properties

Label 2-450-25.9-c1-0-9
Degree $2$
Conductor $450$
Sign $-0.220 + 0.975i$
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  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (1.87 − 1.22i)5-s − 3.26i·7-s + (−0.951 − 0.309i)8-s + (0.107 − 2.23i)10-s + (−1.44 − 1.05i)11-s + (3.38 + 4.65i)13-s + (−2.63 − 1.91i)14-s + (−0.809 + 0.587i)16-s + (−6.26 − 2.03i)17-s + (−1.21 + 3.74i)19-s + (−1.74 − 1.39i)20-s + (−1.70 + 0.552i)22-s + (4.91 − 6.76i)23-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (−0.154 − 0.475i)4-s + (0.836 − 0.548i)5-s − 1.23i·7-s + (−0.336 − 0.109i)8-s + (0.0340 − 0.706i)10-s + (−0.436 − 0.317i)11-s + (0.938 + 1.29i)13-s + (−0.705 − 0.512i)14-s + (−0.202 + 0.146i)16-s + (−1.51 − 0.493i)17-s + (−0.278 + 0.858i)19-s + (−0.389 − 0.313i)20-s + (−0.362 + 0.117i)22-s + (1.02 − 1.41i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15602 - 1.44605i\)
\(L(\frac12)\) \(\approx\) \(1.15602 - 1.44605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 \)
5 \( 1 + (-1.87 + 1.22i)T \)
good7 \( 1 + 3.26iT - 7T^{2} \)
11 \( 1 + (1.44 + 1.05i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.38 - 4.65i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.26 + 2.03i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.21 - 3.74i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.91 + 6.76i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.442 - 1.36i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.99 + 6.15i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.56 - 9.03i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.18 - 2.31i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.18iT - 43T^{2} \)
47 \( 1 + (-6.88 + 2.23i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.47 - 0.478i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.85 - 6.43i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.637 - 0.463i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-7.46 - 2.42i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.12 - 6.55i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.26 - 10.0i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.18 - 3.64i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.37 - 2.07i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.29 + 0.939i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-8.68 + 2.82i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.85987478913232281182279832010, −10.12439298428478929860591153628, −9.150990328408903473656827752875, −8.363763285402486792848582621055, −6.80725823925347408244019094975, −6.15610948835004018717799036872, −4.70427329653420423943749160153, −4.13694982828818824593910750658, −2.49598524934884033806797302694, −1.10839608211035326118421000535, 2.24906097889226931330420998950, 3.26570910727595805260851869726, 4.95121476614419616125579945889, 5.75373396117021537422881917471, 6.45858347016868544937436799368, 7.55194354761330188950446011507, 8.749599169969058300382837460411, 9.251265176788330982376506360433, 10.62841664071569293942601210353, 11.19627154221132515443262515806

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