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Properties

Label 2-450-25.11-c1-0-5
Degree $2$
Conductor $450$
Sign $0.876 + 0.481i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−1.80 + 1.31i)5-s + 2.61·7-s + (−0.309 − 0.951i)8-s + (−0.690 + 2.12i)10-s + (2.92 − 2.12i)11-s + (5.23 + 3.80i)13-s + (2.11 − 1.53i)14-s + (−0.809 − 0.587i)16-s + (−0.381 − 1.17i)17-s + (−1.76 − 5.42i)19-s + (0.690 + 2.12i)20-s + (1.11 − 3.44i)22-s + (3.61 − 2.62i)23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.809 + 0.587i)5-s + 0.989·7-s + (−0.109 − 0.336i)8-s + (−0.218 + 0.672i)10-s + (0.882 − 0.641i)11-s + (1.45 + 1.05i)13-s + (0.566 − 0.411i)14-s + (−0.202 − 0.146i)16-s + (−0.0926 − 0.285i)17-s + (−0.404 − 1.24i)19-s + (0.154 + 0.475i)20-s + (0.238 − 0.733i)22-s + (0.754 − 0.548i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.876 + 0.481i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.876 + 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90561 - 0.489277i\)
\(L(\frac12)\) \(\approx\) \(1.90561 - 0.489277i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 \)
5 \( 1 + (1.80 - 1.31i)T \)
good7 \( 1 - 2.61T + 7T^{2} \)
11 \( 1 + (-2.92 + 2.12i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-5.23 - 3.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.381 + 1.17i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.76 + 5.42i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.61 + 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.61 - 8.05i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.04 - 6.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.47 + 4.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.61 - 3.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.70T + 43T^{2} \)
47 \( 1 + (0.527 - 1.62i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.645 + 1.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.92 + 2.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.472 + 1.45i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.70 - 5.25i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.85 - 2.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.73 - 5.34i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.663 + 2.04i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.85 + 2.07i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.04 - 3.21i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.20051405299337448317526467884, −10.69126987928496592158240311836, −8.977344200700689463302051238167, −8.541974980225482086273975763381, −7.03766812317105271290373538186, −6.47457295041098788075269629925, −4.99037741106377278898739952804, −4.07889290994256452956658891268, −3.11532133991221832367573443558, −1.42617304098453411310418844611, 1.50699297736984822710319012566, 3.60400365938710999774001930088, 4.29606250456022001990513675200, 5.39474710321660758481985524360, 6.36251335656038923613373840579, 7.73890134408509783578342602252, 8.135668108009646514360136479490, 9.067724392237889132371487058262, 10.46761692175684272966888263150, 11.47674635358016795112767840178

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