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Properties

Label 2-1080-120.59-c1-0-72
Degree $2$
Conductor $1080$
Sign $0.373 + 0.927i$
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  + (−1.15 + 0.819i)2-s + (0.657 − 1.88i)4-s + (2.22 − 0.224i)5-s − 2.27·7-s + (0.789 + 2.71i)8-s + (−2.38 + 2.08i)10-s − 3.86i·11-s − 0.654·13-s + (2.61 − 1.86i)14-s + (−3.13 − 2.48i)16-s + 1.82·17-s − 4.12·19-s + (1.03 − 4.34i)20-s + (3.16 + 4.45i)22-s − 7.01i·23-s + ⋯
L(s)  = 1  + (−0.815 + 0.579i)2-s + (0.328 − 0.944i)4-s + (0.994 − 0.100i)5-s − 0.858·7-s + (0.279 + 0.960i)8-s + (−0.752 + 0.658i)10-s − 1.16i·11-s − 0.181·13-s + (0.699 − 0.497i)14-s + (−0.783 − 0.621i)16-s + 0.443·17-s − 0.945·19-s + (0.232 − 0.972i)20-s + (0.675 + 0.950i)22-s − 1.46i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.373 + 0.927i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.373 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8769160140\)
\(L(\frac12)\) \(\approx\) \(0.8769160140\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.819i)T \)
3 \( 1 \)
5 \( 1 + (-2.22 + 0.224i)T \)
good7 \( 1 + 2.27T + 7T^{2} \)
11 \( 1 + 3.86iT - 11T^{2} \)
13 \( 1 + 0.654T + 13T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
19 \( 1 + 4.12T + 19T^{2} \)
23 \( 1 + 7.01iT - 23T^{2} \)
29 \( 1 - 2.64T + 29T^{2} \)
31 \( 1 - 2.38iT - 31T^{2} \)
37 \( 1 + 7.27T + 37T^{2} \)
41 \( 1 + 5.95iT - 41T^{2} \)
43 \( 1 + 9.03iT - 43T^{2} \)
47 \( 1 - 7.13iT - 47T^{2} \)
53 \( 1 + 0.396iT - 53T^{2} \)
59 \( 1 + 14.5iT - 59T^{2} \)
61 \( 1 - 7.03iT - 61T^{2} \)
67 \( 1 + 2.47iT - 67T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 + 9.17iT - 73T^{2} \)
79 \( 1 + 4.35iT - 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 13.4iT - 89T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−9.625458656376959205095616075009, −8.804951996899371608775223500615, −8.387016298595423545674066232141, −7.06112931141421186155545931468, −6.35412185294787550935129452730, −5.82462748664925249273321108114, −4.81647480944833276521693544291, −3.17871112176857843472165500523, −2.03814678083508839677754562156, −0.50290271241766345180085332711, 1.50133167980932486629283612628, 2.48532780532635706432331421631, 3.49329115273434345465767541381, 4.73932829950902088720670403976, 6.01815872642537772454155277852, 6.82456641740906812270204270500, 7.55093967348699032966648618115, 8.663002871638244640631071460499, 9.502553270856530308283085540807, 9.928583749366216422667369237773

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