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Properties

Label 2-1080-5.4-c1-0-8
Degree $2$
Conductor $1080$
Sign $0.994 - 0.100i$
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  + (−2.22 + 0.224i)5-s − i·7-s − 0.449·11-s + 3.89i·13-s − 4.89i·17-s + 5.89·19-s + 4.44i·23-s + (4.89 − i)25-s + 0.449·29-s + 6·31-s + (0.224 + 2.22i)35-s − 0.101i·37-s + 9.34·41-s − 6i·43-s + 4.89i·47-s + ⋯
L(s)  = 1  + (−0.994 + 0.100i)5-s − 0.377i·7-s − 0.135·11-s + 1.08i·13-s − 1.18i·17-s + 1.35·19-s + 0.927i·23-s + (0.979 − 0.200i)25-s + 0.0834·29-s + 1.07·31-s + (0.0379 + 0.376i)35-s − 0.0166i·37-s + 1.45·41-s − 0.914i·43-s + 0.714i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.318836284\)
\(L(\frac12)\) \(\approx\) \(1.318836284\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.22 - 0.224i)T \)
good7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 0.449T + 11T^{2} \)
13 \( 1 - 3.89iT - 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 - 5.89T + 19T^{2} \)
23 \( 1 - 4.44iT - 23T^{2} \)
29 \( 1 - 0.449T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 0.101iT - 37T^{2} \)
41 \( 1 - 9.34T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 4.89iT - 47T^{2} \)
53 \( 1 + 4.44iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 - 8.79T + 61T^{2} \)
67 \( 1 - 14.7iT - 67T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + 3.89iT - 73T^{2} \)
79 \( 1 - 3.89T + 79T^{2} \)
83 \( 1 - 7.55iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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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

−9.763403724311856000301787135332, −9.163407510001933213351483462551, −8.107239784601655893503006548854, −7.33030441075050857515636205375, −6.85116609336243496825050146052, −5.50581794631919615383208040384, −4.55732187720825359557970819878, −3.72285226185615309856788897395, −2.66933529378848110097223819157, −0.928965721584992972493163018449, 0.867057937735633417282932925047, 2.66163040182725667415818103655, 3.59073139995616428503221657645, 4.61215967172612307793336079584, 5.55751772577954270718501211801, 6.48701999791385397793303490252, 7.63672542566775209068636803105, 8.100066048369795637886686014650, 8.889457253960636705652971547551, 9.939764983358776603369532663247

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