Properties

Label 2-1440-9.7-c1-0-0
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: $\chi_{1440} (961, \cdot )$
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

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

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