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Properties

Label 2-1080-8.5-c1-0-45
Degree $2$
Conductor $1080$
Sign $-0.127 + 0.991i$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
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  + (−1.19 − 0.758i)2-s + (0.849 + 1.81i)4-s − i·5-s + 4.63·7-s + (0.359 − 2.80i)8-s + (−0.758 + 1.19i)10-s − 2.39i·11-s − 1.95i·13-s + (−5.52 − 3.51i)14-s + (−2.55 + 3.07i)16-s − 4.76·17-s − 6.29i·19-s + (1.81 − 0.849i)20-s + (−1.81 + 2.85i)22-s + 5.28·23-s + ⋯
L(s)  = 1  + (−0.843 − 0.536i)2-s + (0.424 + 0.905i)4-s − 0.447i·5-s + 1.75·7-s + (0.127 − 0.991i)8-s + (−0.239 + 0.377i)10-s − 0.722i·11-s − 0.541i·13-s + (−1.47 − 0.939i)14-s + (−0.639 + 0.768i)16-s − 1.15·17-s − 1.44i·19-s + (0.404 − 0.189i)20-s + (−0.387 + 0.609i)22-s + 1.10·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.127 + 0.991i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.127 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210848462\)
\(L(\frac12)\) \(\approx\) \(1.210848462\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.758i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 - 4.63T + 7T^{2} \)
11 \( 1 + 2.39iT - 11T^{2} \)
13 \( 1 + 1.95iT - 13T^{2} \)
17 \( 1 + 4.76T + 17T^{2} \)
19 \( 1 + 6.29iT - 19T^{2} \)
23 \( 1 - 5.28T + 23T^{2} \)
29 \( 1 - 0.201iT - 29T^{2} \)
31 \( 1 + 3.26T + 31T^{2} \)
37 \( 1 - 1.43iT - 37T^{2} \)
41 \( 1 - 8.53T + 41T^{2} \)
43 \( 1 - 9.10iT - 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 3.03iT - 53T^{2} \)
59 \( 1 + 2.75iT - 59T^{2} \)
61 \( 1 + 8.24iT - 61T^{2} \)
67 \( 1 + 12.0iT - 67T^{2} \)
71 \( 1 - 0.514T + 71T^{2} \)
73 \( 1 + 4.03T + 73T^{2} \)
79 \( 1 + 1.43T + 79T^{2} \)
83 \( 1 - 7.66iT - 83T^{2} \)
89 \( 1 - 9.10T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−9.447833273126174754785603441690, −8.816955591582811062798127579514, −8.205486452294042943810671421013, −7.51969140677641997026302986740, −6.48440051903894549086353069098, −5.08548893488684513352027295049, −4.45992976215088839450607275976, −3.04585917439276562616268580944, −1.91625158713855627122080784565, −0.76068859733356433033034398559, 1.50261505262228138977820048223, 2.28139063451276209987546408041, 4.20503102691142266303029364683, 5.04219246841296529925737286431, 5.98468239095677748617888883648, 7.09563228298125504358697246537, 7.54417839926848276572390278703, 8.475188924574582366225212401180, 9.061786401015388232612210909947, 10.10250752820035258603863484360

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