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Properties

Label 2-450-15.8-c3-0-11
Degree $2$
Conductor $450$
Sign $-0.749 + 0.662i$
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 + (−21.9 + 21.9i)7-s + (5.65 − 5.65i)8-s + 64.9i·11-s + (−13.0 − 13.0i)13-s + 61.9·14-s − 16.0·16-s + (−5.53 − 5.53i)17-s − 103. i·19-s + (91.8 − 91.8i)22-s + (19.7 − 19.7i)23-s + 37.0i·26-s + (−87.6 − 87.6i)28-s − 184.·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−1.18 + 1.18i)7-s + (0.250 − 0.250i)8-s + 1.77i·11-s + (−0.279 − 0.279i)13-s + 1.18·14-s − 0.250·16-s + (−0.0789 − 0.0789i)17-s − 1.25i·19-s + (0.889 − 0.889i)22-s + (0.179 − 0.179i)23-s + 0.279i·26-s + (−0.591 − 0.591i)28-s − 1.18·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2385079276\)
\(L(\frac12)\) \(\approx\) \(0.2385079276\)
\(L(\frac{5}{2})\) 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 \( 1 + (21.9 - 21.9i)T - 343iT^{2} \)
11 \( 1 - 64.9iT - 1.33e3T^{2} \)
13 \( 1 + (13.0 + 13.0i)T + 2.19e3iT^{2} \)
17 \( 1 + (5.53 + 5.53i)T + 4.91e3iT^{2} \)
19 \( 1 + 103. iT - 6.85e3T^{2} \)
23 \( 1 + (-19.7 + 19.7i)T - 1.21e4iT^{2} \)
29 \( 1 + 184.T + 2.43e4T^{2} \)
31 \( 1 - 19.8T + 2.97e4T^{2} \)
37 \( 1 + (-254. + 254. i)T - 5.06e4iT^{2} \)
41 \( 1 + 63.1iT - 6.89e4T^{2} \)
43 \( 1 + (104. + 104. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-185. - 185. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-259. + 259. i)T - 1.48e5iT^{2} \)
59 \( 1 - 255.T + 2.05e5T^{2} \)
61 \( 1 + 894.T + 2.26e5T^{2} \)
67 \( 1 + (-100. + 100. i)T - 3.00e5iT^{2} \)
71 \( 1 - 491. iT - 3.57e5T^{2} \)
73 \( 1 + (603. + 603. i)T + 3.89e5iT^{2} \)
79 \( 1 + 698. iT - 4.93e5T^{2} \)
83 \( 1 + (707. - 707. i)T - 5.71e5iT^{2} \)
89 \( 1 + 389.T + 7.04e5T^{2} \)
97 \( 1 + (-29.2 + 29.2i)T - 9.12e5iT^{2} \)
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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

−10.10560585958871364757344169624, −9.415474467919973710936588526896, −8.941188405647109442274306099219, −7.53368034724673102423798004394, −6.79148500608418750853528077645, −5.56360540702168653045085313727, −4.34408140153248248918156243376, −2.89390104904532650768433436141, −2.09820585039039791271183414620, −0.10278103178573784874208114481, 1.08525100897283825383376187990, 3.13145440535453227653465138875, 4.08644594792638226702460351917, 5.72336326319954543381780669630, 6.38755274940457145049492732436, 7.36633496034727053116725081921, 8.230199230896381010373642989850, 9.248836239576760186604020684750, 10.05620333698242812894057662816, 10.77227524311369812458329520075

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