Properties

Label 2-1080-8.5-c1-0-46
Degree $2$
Conductor $1080$
Sign $0.818 + 0.574i$
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  + (0.444 + 1.34i)2-s + (−1.60 + 1.19i)4-s + i·5-s − 1.61·7-s + (−2.31 − 1.62i)8-s + (−1.34 + 0.444i)10-s − 2.72i·11-s − 6.12i·13-s + (−0.718 − 2.16i)14-s + (1.15 − 3.83i)16-s − 4.31·17-s − 2.87i·19-s + (−1.19 − 1.60i)20-s + (3.65 − 1.21i)22-s + 7.44·23-s + ⋯
L(s)  = 1  + (0.314 + 0.949i)2-s + (−0.802 + 0.596i)4-s + 0.447i·5-s − 0.610·7-s + (−0.818 − 0.574i)8-s + (−0.424 + 0.140i)10-s − 0.821i·11-s − 1.70i·13-s + (−0.192 − 0.579i)14-s + (0.287 − 0.957i)16-s − 1.04·17-s − 0.660i·19-s + (−0.266 − 0.358i)20-s + (0.779 − 0.258i)22-s + 1.55·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.818 + 0.574i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.818 + 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9067631595\)
\(L(\frac12)\) \(\approx\) \(0.9067631595\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.444 - 1.34i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 + 2.72iT - 11T^{2} \)
13 \( 1 + 6.12iT - 13T^{2} \)
17 \( 1 + 4.31T + 17T^{2} \)
19 \( 1 + 2.87iT - 19T^{2} \)
23 \( 1 - 7.44T + 23T^{2} \)
29 \( 1 + 1.43iT - 29T^{2} \)
31 \( 1 + 9.17T + 31T^{2} \)
37 \( 1 - 9.26iT - 37T^{2} \)
41 \( 1 - 6.77T + 41T^{2} \)
43 \( 1 + 7.56iT - 43T^{2} \)
47 \( 1 + 0.642T + 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 14.5iT - 59T^{2} \)
61 \( 1 + 9.00iT - 61T^{2} \)
67 \( 1 + 6.55iT - 67T^{2} \)
71 \( 1 - 3.13T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 1.53T + 79T^{2} \)
83 \( 1 + 8.44iT - 83T^{2} \)
89 \( 1 - 7.56T + 89T^{2} \)
97 \( 1 + 1.11T + 97T^{2} \)
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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

−9.543203131238446046207481856560, −8.862873849436078050359670966302, −8.024146082453502464767152059912, −7.16782731107296722963154935393, −6.44328043784774708055878203923, −5.64207700530444082083675013756, −4.80481866279846348365407465775, −3.45899502385583087107818919499, −2.89858906030674403038597874693, −0.36745660121073038551808808024, 1.51341812583614712886649484217, 2.49906978373295543014350492816, 3.86755218295112337056968054613, 4.46784895875091469649386491297, 5.44979177749977144531829255137, 6.53397763317718199848404321846, 7.35927378128160724461961270639, 8.942319660774101017319414038361, 9.097738834351311417743375305034, 9.905886486021091866875210843433

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