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Properties

Label 2-450-15.2-c3-0-16
Degree $2$
Conductor $450$
Sign $-0.884 + 0.465i$
Analytic cond. $26.5508$
Root an. cond. $5.15275$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 1.41i)2-s − 4.00i·4-s + (0.123 + 0.123i)7-s + (−5.65 − 5.65i)8-s − 4.24i·11-s + (−5.62 + 5.62i)13-s + 0.349·14-s − 16.0·16-s + (34.4 − 34.4i)17-s − 110. i·19-s + (−6 − 6i)22-s + (−111. − 111. i)23-s + 15.9i·26-s + (0.494 − 0.494i)28-s − 53.0·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.00668 + 0.00668i)7-s + (−0.250 − 0.250i)8-s − 0.116i·11-s + (−0.120 + 0.120i)13-s + 0.00668·14-s − 0.250·16-s + (0.491 − 0.491i)17-s − 1.33i·19-s + (−0.0581 − 0.0581i)22-s + (−1.01 − 1.01i)23-s + 0.120i·26-s + (0.00334 − 0.00334i)28-s − 0.339·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.465i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.884 + 0.465i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ -0.884 + 0.465i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.609216242\)
\(L(\frac12)\) \(\approx\) \(1.609216242\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 1.41i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.123 - 0.123i)T + 343iT^{2} \)
11 \( 1 + 4.24iT - 1.33e3T^{2} \)
13 \( 1 + (5.62 - 5.62i)T - 2.19e3iT^{2} \)
17 \( 1 + (-34.4 + 34.4i)T - 4.91e3iT^{2} \)
19 \( 1 + 110. iT - 6.85e3T^{2} \)
23 \( 1 + (111. + 111. i)T + 1.21e4iT^{2} \)
29 \( 1 + 53.0T + 2.43e4T^{2} \)
31 \( 1 + 201.T + 2.97e4T^{2} \)
37 \( 1 + (-169. - 169. i)T + 5.06e4iT^{2} \)
41 \( 1 + 371. iT - 6.89e4T^{2} \)
43 \( 1 + (202. - 202. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-106. + 106. i)T - 1.03e5iT^{2} \)
53 \( 1 + (243. + 243. i)T + 1.48e5iT^{2} \)
59 \( 1 - 57.2T + 2.05e5T^{2} \)
61 \( 1 + 609.T + 2.26e5T^{2} \)
67 \( 1 + (-340. - 340. i)T + 3.00e5iT^{2} \)
71 \( 1 + 990. iT - 3.57e5T^{2} \)
73 \( 1 + (847. - 847. i)T - 3.89e5iT^{2} \)
79 \( 1 + 436. iT - 4.93e5T^{2} \)
83 \( 1 + (-127. - 127. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.23e3T + 7.04e5T^{2} \)
97 \( 1 + (202. + 202. i)T + 9.12e5iT^{2} \)
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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

−10.41474184251708071224471516125, −9.539204492718314921880652909061, −8.632844076755534095175410425542, −7.42404274156523324351664537752, −6.42937940832316577479255144956, −5.33686208297135469062357037970, −4.39784389491482543955781306098, −3.20642122320252379283844185867, −2.04622799233086758422862481359, −0.41828337039557484367678150818, 1.71749762031585624425003573097, 3.32430490906886233459915689278, 4.24728970125173622566073808272, 5.55064276291878092559534659562, 6.15905075771877659914567592311, 7.50712364627878981743245060979, 7.995543199332498259368995075138, 9.233995346769896416383392378887, 10.11148395312107611593686461138, 11.15245435745206219503144868551

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