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Properties

Label 2-1440-72.11-c1-0-17
Degree $2$
Conductor $1440$
Sign $0.874 - 0.485i$
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  + (−1.72 + 0.199i)3-s + (−0.5 − 0.866i)5-s + (3.97 + 2.29i)7-s + (2.92 − 0.686i)9-s + (4.08 + 2.36i)11-s + (1.87 − 1.08i)13-s + (1.03 + 1.39i)15-s − 1.23i·17-s − 4.35·19-s + (−7.29 − 3.15i)21-s + (−0.117 − 0.204i)23-s + (−0.499 + 0.866i)25-s + (−4.88 + 1.76i)27-s + (2.84 − 4.92i)29-s + (−4.06 + 2.34i)31-s + ⋯
L(s)  = 1  + (−0.993 + 0.115i)3-s + (−0.223 − 0.387i)5-s + (1.50 + 0.866i)7-s + (0.973 − 0.228i)9-s + (1.23 + 0.711i)11-s + (0.518 − 0.299i)13-s + (0.266 + 0.358i)15-s − 0.299i·17-s − 1.00·19-s + (−1.59 − 0.688i)21-s + (−0.0245 − 0.0425i)23-s + (−0.0999 + 0.173i)25-s + (−0.940 + 0.339i)27-s + (0.527 − 0.914i)29-s + (−0.730 + 0.421i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.517404823\)
\(L(\frac12)\) \(\approx\) \(1.517404823\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.199i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-3.97 - 2.29i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.08 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.87 + 1.08i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.23iT - 17T^{2} \)
19 \( 1 + 4.35T + 19T^{2} \)
23 \( 1 + (0.117 + 0.204i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.84 + 4.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.06 - 2.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.91iT - 37T^{2} \)
41 \( 1 + (-6.34 + 3.66i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.62 + 4.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.05 - 1.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.92T + 53T^{2} \)
59 \( 1 + (-10.1 + 5.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.27 - 3.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.51 - 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.851T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (10.0 + 5.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.35 - 1.35i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 + (0.816 - 1.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.580694637244682615143297910779, −8.739660348360551739259559763720, −8.103745515365731771934886724551, −7.06119584270909152874827104710, −6.21222525291817561181432397719, −5.36600202534373743149841621321, −4.62345101948994773752161109950, −3.98198169681963640414037596711, −2.10218722334072107071203349344, −1.13722265372138897474882445088, 0.909668488140401883445067823332, 1.86396650213634766175567263544, 3.85030086580523998418802551302, 4.26013901738049471408745095564, 5.32785239823643233922280229101, 6.28825581546627175759694109488, 6.90938443149807047728254794106, 7.78100646218888863543768099275, 8.525943139645162074186198092341, 9.527648974788473846736831522939

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