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Properties

Label 2-1440-5.4-c1-0-10
Degree $2$
Conductor $1440$
Sign $-i$
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  + 2.23·5-s + 5.23i·7-s + 7.70i·23-s + 5.00·25-s − 6·29-s + 11.7i·35-s − 4.47·41-s − 6.76i·43-s − 0.291i·47-s − 20.4·49-s + 13.4·61-s + 14.1i·67-s − 4.29i·83-s + 6·89-s + 18·101-s + ⋯
L(s)  = 1  + 0.999·5-s + 1.97i·7-s + 1.60i·23-s + 1.00·25-s − 1.11·29-s + 1.97i·35-s − 0.698·41-s − 1.03i·43-s − 0.0425i·47-s − 2.91·49-s + 1.71·61-s + 1.73i·67-s − 0.471i·83-s + 0.635·89-s + 1.79·101-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.820258837\)
\(L(\frac12)\) \(\approx\) \(1.820258837\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
good7 \( 1 - 5.23iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 7.70iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 6.76iT - 43T^{2} \)
47 \( 1 + 0.291iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.29iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.536764657580066590245870416674, −9.046964183495618368165232680894, −8.379605146069918487673359172955, −7.25055136705189063964870704367, −6.22429558371732792525081163101, −5.54407433023342189273055146191, −5.13790483574158477424427236815, −3.52220768032087563219414739347, −2.45767430073303107360858235149, −1.72793885768563478206225408784, 0.72238767160405768456231364357, 1.92637502767769634421162904253, 3.26955170548487771005438957371, 4.26982624030651213656448852857, 5.03719330713400564365034910379, 6.23787131725952708877832366772, 6.83244411405491439020095373372, 7.61671613390071271585061664043, 8.523451053519244889401467553152, 9.548689605235976676016066996164

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