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Properties

Label 2-1440-9.4-c1-0-47
Degree $2$
Conductor $1440$
Sign $-0.989 - 0.142i$
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  + (1.40 − 1.01i)3-s + (−0.5 + 0.866i)5-s + (−1.40 − 2.42i)7-s + (0.936 − 2.85i)9-s + (−1.22 − 2.12i)11-s + (−2.43 + 4.22i)13-s + (0.178 + 1.72i)15-s − 6.87·17-s − 7.34·19-s + (−4.43 − 1.98i)21-s + (−1.40 + 2.42i)23-s + (−0.499 − 0.866i)25-s + (−1.58 − 4.94i)27-s + (3.37 + 5.84i)29-s + (1.93 − 3.35i)31-s + ⋯
L(s)  = 1  + (0.809 − 0.586i)3-s + (−0.223 + 0.387i)5-s + (−0.530 − 0.918i)7-s + (0.312 − 0.950i)9-s + (−0.369 − 0.639i)11-s + (−0.675 + 1.17i)13-s + (0.0460 + 0.444i)15-s − 1.66·17-s − 1.68·19-s + (−0.968 − 0.432i)21-s + (−0.292 + 0.506i)23-s + (−0.0999 − 0.173i)25-s + (−0.304 − 0.952i)27-s + (0.626 + 1.08i)29-s + (0.347 − 0.602i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.989 - 0.142i$
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: \((2,\ 1440,\ (\ :1/2),\ -0.989 - 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4542293104\)
\(L(\frac12)\) \(\approx\) \(0.4542293104\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 1.01i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (1.40 + 2.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.22 + 2.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.43 - 4.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.87T + 17T^{2} \)
19 \( 1 + 7.34T + 19T^{2} \)
23 \( 1 + (1.40 - 2.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.37 - 5.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.93 + 3.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.872T + 37T^{2} \)
41 \( 1 + (-1.06 + 1.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.22 - 2.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.98 - 5.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (-2.80 + 4.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.93 + 5.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.88 + 13.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.16T + 71T^{2} \)
73 \( 1 + 9.74T + 73T^{2} \)
79 \( 1 + (6.32 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.72 - 13.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + (-0.872 - 1.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.014572309551755639712131138958, −8.319522935282739390999502980736, −7.42016516950919119102206511606, −6.69425209580870053225945624681, −6.31827632816109110693418927120, −4.52986857388821748770809719147, −3.92054567331883457243546804739, −2.81700044605552682955604445337, −1.90909187132106449779668422261, −0.14603365943198417206469598043, 2.30915337811698363747287333715, 2.68122863968921272943040881154, 4.15599473555772189571749850436, 4.69258458202111760596282192335, 5.73308004794111548551015687261, 6.74953498908471309517755028921, 7.77371293491427309993518147992, 8.638021138713584031936865916294, 8.872814435144389668436827230716, 10.04881785648851403742562183757

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