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Properties

Label 2-450-25.9-c1-0-0
Degree $2$
Conductor $450$
Sign $-0.559 - 0.829i$
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 + (−1.86 − 1.23i)5-s + 2.70i·7-s + (0.951 + 0.309i)8-s + (2.09 − 0.786i)10-s + (4.54 + 3.30i)11-s + (−2.84 − 3.91i)13-s + (−2.19 − 1.59i)14-s + (−0.809 + 0.587i)16-s + (−0.994 − 0.323i)17-s + (−2.59 + 7.97i)19-s + (−0.593 + 2.15i)20-s + (−5.34 + 1.73i)22-s + (−3.11 + 4.28i)23-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.154 − 0.475i)4-s + (−0.834 − 0.550i)5-s + 1.02i·7-s + (0.336 + 0.109i)8-s + (0.661 − 0.248i)10-s + (1.37 + 0.996i)11-s + (−0.789 − 1.08i)13-s + (−0.585 − 0.425i)14-s + (−0.202 + 0.146i)16-s + (−0.241 − 0.0783i)17-s + (−0.594 + 1.82i)19-s + (−0.132 + 0.482i)20-s + (−1.13 + 0.370i)22-s + (−0.648 + 0.893i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.559 - 0.829i$
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: \((2,\ 450,\ (\ :1/2),\ -0.559 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344595 + 0.648141i\)
\(L(\frac12)\) \(\approx\) \(0.344595 + 0.648141i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 1.23i)T \)
good7 \( 1 - 2.70iT - 7T^{2} \)
11 \( 1 + (-4.54 - 3.30i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.84 + 3.91i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.994 + 0.323i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.59 - 7.97i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.11 - 4.28i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.29 - 3.98i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.72 - 5.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.75 - 2.42i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.27 - 0.927i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.29iT - 43T^{2} \)
47 \( 1 + (2.92 - 0.949i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.87 - 1.90i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-11.4 + 8.33i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.218 + 0.159i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-6.00 - 1.94i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.40 - 4.31i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.79 + 6.60i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.85 + 11.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.127 + 0.0415i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (9.53 + 6.93i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.51 + 0.815i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.58679749897819167255142937874, −10.22906676016153588271376878217, −9.455906790663486245389646878200, −8.575476424894188037675542863743, −7.87721526272314052696907100891, −6.90954405781986517089035560047, −5.76510418714209391826273847514, −4.82919013651799573475109643518, −3.62430361677375932830243050678, −1.69571486238291291327361680738, 0.54072288057708896209974735271, 2.48956288855676665633668244548, 3.93220617985790401926570611851, 4.37621968027982225533715269525, 6.54909356768337225443508623296, 7.01956405561389794757411441436, 8.162894844865586476786828973822, 9.043722036431206094604237027359, 9.939673871864419096091755949066, 11.14673585603340117151216229008

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