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Properties

Label 2-1080-15.14-c0-0-3
Degree $2$
Conductor $1080$
Sign $0.707 + 0.707i$
Analytic cond. $0.538990$
Root an. cond. $0.734159$
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  + (0.707 − 0.707i)5-s − i·7-s + 1.41i·11-s − i·13-s + 19-s − 1.41·23-s − 1.00i·25-s − 1.41i·29-s + (−0.707 − 0.707i)35-s + i·37-s + 1.41i·41-s + 1.41·53-s + (1.00 + 1.00i)55-s − 61-s + (−0.707 − 0.707i)65-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s − i·7-s + 1.41i·11-s − i·13-s + 19-s − 1.41·23-s − 1.00i·25-s − 1.41i·29-s + (−0.707 − 0.707i)35-s + i·37-s + 1.41i·41-s + 1.41·53-s + (1.00 + 1.00i)55-s − 61-s + (−0.707 − 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.538990\)
Root analytic conductor: \(0.734159\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152337091\)
\(L(\frac12)\) \(\approx\) \(1.152337091\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.980707808180055419181492890554, −9.445095282307065858004135029279, −8.132861237411487783508891519334, −7.64518874221820060885310010851, −6.60789092096609919081771770500, −5.64874430044957381201714166989, −4.74684725050412620870627595115, −3.96030308687788534337122033046, −2.48326890417605766335319548930, −1.21180907453429959613752507040, 1.82798030261596932416146894108, 2.85907310983920148881395774872, 3.81595426820235872111729045947, 5.39488487704278120896722878890, 5.84197774650761826389011109558, 6.70516456762575239617454662887, 7.65522180110661297687714086515, 8.876202610381499963264798681529, 9.127520693697156653788706545636, 10.22676620052850779604026883059

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