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Properties

Label 2-1440-60.59-c1-0-16
Degree $2$
Conductor $1440$
Sign $0.999 - 0.0159i$
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  + (2.20 + 0.342i)5-s + 2.64·7-s + 3.00·11-s + 0.640i·13-s + 0.685·17-s − 5.28i·19-s − 2.27i·23-s + (4.76 + 1.51i)25-s + 8.15i·29-s − 2.96i·31-s + (5.83 + 0.905i)35-s − 1.60i·37-s + 7.42i·41-s − 11.2·43-s − 4.19i·47-s + ⋯
L(s)  = 1  + (0.988 + 0.153i)5-s + 0.997·7-s + 0.906·11-s + 0.177i·13-s + 0.166·17-s − 1.21i·19-s − 0.474i·23-s + (0.952 + 0.303i)25-s + 1.51i·29-s − 0.533i·31-s + (0.986 + 0.152i)35-s − 0.264i·37-s + 1.15i·41-s − 1.71·43-s − 0.612i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.999 - 0.0159i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.999 - 0.0159i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.417935219\)
\(L(\frac12)\) \(\approx\) \(2.417935219\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-2.20 - 0.342i)T \)
good7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 - 3.00T + 11T^{2} \)
13 \( 1 - 0.640iT - 13T^{2} \)
17 \( 1 - 0.685T + 17T^{2} \)
19 \( 1 + 5.28iT - 19T^{2} \)
23 \( 1 + 2.27iT - 23T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 + 2.96iT - 31T^{2} \)
37 \( 1 + 1.60iT - 37T^{2} \)
41 \( 1 - 7.42iT - 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + 4.19iT - 47T^{2} \)
53 \( 1 + 9.60T + 53T^{2} \)
59 \( 1 - 7.20T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 8.49T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + 14.2iT - 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 - 13.1iT - 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 8.31iT - 97T^{2} \)
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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

−9.433333597398175488910870332785, −8.857350059312166274725128018779, −8.022694874067574737389182796472, −6.88186731663065709118883602550, −6.42281592128476187321933816126, −5.21833092085582492631506997927, −4.71482126403620168732566422765, −3.40254450484771085711929949319, −2.19657517208846969631776006082, −1.26094736306154371189773463066, 1.30276346639425110720743680454, 2.08482843651040246736130278604, 3.50535006539908865173175696114, 4.55222505078480460876240497318, 5.44611569819587789865499195625, 6.13094125042360148938237892230, 7.04877215880603822462349646369, 8.111122052873353495938763684019, 8.632317861369513607044128951998, 9.715763329010054806821232962004

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