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Properties

Label 2-1440-9.4-c1-0-34
Degree $2$
Conductor $1440$
Sign $-0.0315 + 0.999i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
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  + (−0.178 + 1.72i)3-s + (−0.5 + 0.866i)5-s + (0.178 + 0.308i)7-s + (−2.93 − 0.614i)9-s + (−1.22 − 2.12i)11-s + (1.43 − 2.48i)13-s + (−1.40 − 1.01i)15-s + 0.872·17-s − 7.34·19-s + (−0.563 + 0.252i)21-s + (0.178 − 0.308i)23-s + (−0.499 − 0.866i)25-s + (1.58 − 4.94i)27-s + (−4.37 − 7.57i)29-s + (−4.38 + 7.59i)31-s + ⋯
L(s)  = 1  + (−0.102 + 0.994i)3-s + (−0.223 + 0.387i)5-s + (0.0673 + 0.116i)7-s + (−0.978 − 0.204i)9-s + (−0.369 − 0.639i)11-s + (0.398 − 0.690i)13-s + (−0.362 − 0.262i)15-s + 0.211·17-s − 1.68·19-s + (−0.122 + 0.0549i)21-s + (0.0371 − 0.0643i)23-s + (−0.0999 − 0.173i)25-s + (0.304 − 0.952i)27-s + (−0.812 − 1.40i)29-s + (−0.787 + 1.36i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0315 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0315 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.0315 + 0.999i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.0315 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4190259819\)
\(L(\frac12)\) \(\approx\) \(0.4190259819\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.178 - 1.72i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.178 - 0.308i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.22 + 2.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.43 + 2.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.872T + 17T^{2} \)
19 \( 1 + 7.34T + 19T^{2} \)
23 \( 1 + (-0.178 + 0.308i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.37 + 7.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.38 - 7.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
41 \( 1 + (-4.93 + 8.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.22 - 2.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.75 + 3.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (0.356 - 0.617i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.936 - 1.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.13 + 5.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.16T + 71T^{2} \)
73 \( 1 - 5.74T + 73T^{2} \)
79 \( 1 + (-6.32 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.50 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.25T + 89T^{2} \)
97 \( 1 + (6.87 + 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.301458155249739341323436004966, −8.556893889820362864801555045064, −7.974311673424769145362228759009, −6.76587340878673584536599890030, −5.85093625153744558959719460229, −5.21813736860699248451779392925, −4.04873735305136271954241857959, −3.41417029521922794731860479887, −2.30409148068877064700445689992, −0.16505397072170838678348247836, 1.46870259109114233442460401339, 2.35915732856958396086625064798, 3.75987814417553559238394910258, 4.74366241794064799909074620839, 5.73315027136404712680996856350, 6.57682330861082088206524158348, 7.32477560537067771117778152073, 8.051814647267908410450828663106, 8.809899040626298709550162087545, 9.541127817798450904399753536289

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