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Properties

Label 2-450-25.4-c1-0-2
Degree $2$
Conductor $450$
Sign $-0.208 - 0.978i$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−2.23 − 0.0466i)5-s + 3.52i·7-s + (0.587 + 0.809i)8-s + (−2.11 − 0.735i)10-s + (−1.62 + 4.99i)11-s + (−0.588 + 0.191i)13-s + (−1.08 + 3.34i)14-s + (0.309 + 0.951i)16-s + (−2.02 − 2.78i)17-s + (1.83 − 1.33i)19-s + (−1.78 − 1.35i)20-s + (−3.08 + 4.25i)22-s + (8.51 + 2.76i)23-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.404 + 0.293i)4-s + (−0.999 − 0.0208i)5-s + 1.33i·7-s + (0.207 + 0.286i)8-s + (−0.667 − 0.232i)10-s + (−0.489 + 1.50i)11-s + (−0.163 + 0.0530i)13-s + (−0.290 + 0.895i)14-s + (0.0772 + 0.237i)16-s + (−0.491 − 0.676i)17-s + (0.422 − 0.306i)19-s + (−0.398 − 0.302i)20-s + (−0.658 + 0.906i)22-s + (1.77 + 0.576i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.208 - 0.978i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.208 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.969456 + 1.19742i\)
\(L(\frac12)\) \(\approx\) \(0.969456 + 1.19742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 \)
5 \( 1 + (2.23 + 0.0466i)T \)
good7 \( 1 - 3.52iT - 7T^{2} \)
11 \( 1 + (1.62 - 4.99i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.588 - 0.191i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.02 + 2.78i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.83 + 1.33i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-8.51 - 2.76i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.16 - 1.57i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (7.90 - 5.74i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.952 - 0.309i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.94 + 5.98i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.51iT - 43T^{2} \)
47 \( 1 + (-6.27 + 8.63i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.325 - 0.447i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.0861 - 0.265i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.13 + 3.50i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.00 + 9.63i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.84 - 2.79i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-9.35 - 3.03i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.27 - 3.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.56 - 2.15i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.13 - 12.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.56 - 3.53i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.57435682005150765934834226716, −10.73442196095117412147845526185, −9.335225266319681354046349375106, −8.642627534718507318721378053933, −7.33055901125390993446425841880, −6.96164729730794461630356024606, −5.25142881281840208783080894552, −4.88457312315378130909676796926, −3.41192713819974300584681897356, −2.28741300768822514503708549684, 0.793516707318753513436134825722, 3.03743513789464361714261164363, 3.86600225106217591947179762590, 4.80278677543492500750383298004, 6.08140783521985676710330766784, 7.17479542574435868002265670771, 7.897334369572400576352133207334, 8.940199072394315081561906577161, 10.42016630006285267204630813135, 10.97398613552715281277999519446

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