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Properties

Label 2-1440-160.99-c0-0-0
Degree $2$
Conductor $1440$
Sign $0.831 - 0.555i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (−0.382 − 0.923i)8-s + 10-s + i·16-s + 0.765·17-s + (−0.707 − 0.292i)19-s + (−0.923 − 0.382i)20-s + (1.30 + 1.30i)23-s + (0.707 − 0.707i)25-s + 1.41i·31-s + (0.382 − 0.923i)32-s + (−0.707 − 0.292i)34-s + (0.541 + 0.541i)38-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (−0.382 − 0.923i)8-s + 10-s + i·16-s + 0.765·17-s + (−0.707 − 0.292i)19-s + (−0.923 − 0.382i)20-s + (1.30 + 1.30i)23-s + (0.707 − 0.707i)25-s + 1.41i·31-s + (0.382 − 0.923i)32-s + (−0.707 − 0.292i)34-s + (0.541 + 0.541i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5801096308\)
\(L(\frac12)\) \(\approx\) \(0.5801096308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
3 \( 1 \)
5 \( 1 + (0.923 - 0.382i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T^{2} \)
17 \( 1 - 0.765T + T^{2} \)
19 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 - 0.765iT - T^{2} \)
53 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.707 + 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−9.809428032427190763223473541213, −8.924091662319262042546805342844, −8.323572696234754749831455531166, −7.35203890280104264870497430845, −7.04601133022641415034082796513, −5.86213950962041211704851548290, −4.54529475350383657685409126796, −3.47458461394986412043479748799, −2.79152695111617998007725260282, −1.24295296608420281820331703325, 0.74949933951844820024991701889, 2.29070999588952327713000647588, 3.58656145403288001570791057678, 4.73269438085578848559333386171, 5.61491885029944216755455745206, 6.69287766483849275508376101269, 7.30567991574776216094714213228, 8.318308960983510620624331615397, 8.524120343653076847663956231049, 9.575191089631453016172165642121

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