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Properties

Label 2-1080-8.5-c1-0-58
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

\(L(1)\) \(\approx\) \(3.158840499\)
\(L(\frac12)\) \(\approx\) \(3.158840499\)
\(L(\frac{3}{2})\) 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

−10.01568974001294674156992862582, −8.786943936731730272336242851992, −8.089548338956543494451124931837, −7.20253771794455594467586463935, −5.71076034072429009786537585462, −5.48247641159419411309260205603, −4.32484171233485554036161391366, −3.64832389687858245050807970230, −2.06868364794722566724886685659, −1.26042648908923419880770235415, 1.78673310183967753539713948844, 2.90265091937747401724823317348, 4.14979000871947086510623183167, 4.94308146757364237894638953051, 5.60387683617023401045349384469, 6.73043439127346317808612053598, 7.68987344897205602091373729178, 7.937499094313103421964127600219, 9.031584210847852239671977198065, 10.27926957549603898844808512459

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