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Properties

Label 2-1080-120.59-c1-0-19
Degree $2$
Conductor $1080$
Sign $0.395 - 0.918i$
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  + (−0.535 − 1.30i)2-s + (−1.42 + 1.40i)4-s + 2.23i·5-s + (2.59 + 1.11i)8-s + (2.92 − 1.19i)10-s + (0.0729 − 3.99i)16-s + 7.33·17-s − 8.70·19-s + (−3.13 − 3.19i)20-s + 1.47i·23-s − 5.00·25-s + 10.8i·31-s + (−5.27 + 2.04i)32-s + (−3.92 − 9.60i)34-s + (4.66 + 11.3i)38-s + ⋯
L(s)  = 1  + (−0.378 − 0.925i)2-s + (−0.713 + 0.700i)4-s + 0.999i·5-s + (0.918 + 0.395i)8-s + (0.925 − 0.378i)10-s + (0.0182 − 0.999i)16-s + 1.77·17-s − 1.99·19-s + (−0.700 − 0.713i)20-s + 0.306i·23-s − 1.00·25-s + 1.93i·31-s + (−0.932 + 0.361i)32-s + (−0.673 − 1.64i)34-s + (0.756 + 1.84i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.395 - 0.918i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8622943572\)
\(L(\frac12)\) \(\approx\) \(0.8622943572\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.535 + 1.30i)T \)
3 \( 1 \)
5 \( 1 - 2.23iT \)
good7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.33T + 17T^{2} \)
19 \( 1 + 8.70T + 19T^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 10.8iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 - 14.2iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 6.01iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 9.98iT - 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.24304716953014254747739118871, −9.426508901479061622093546900654, −8.437313984258015293826175428024, −7.73106651518823137620929830475, −6.82498889432606329707800456626, −5.77642154599179172517843070708, −4.54278367022667073034991405219, −3.50343833747253494790540842033, −2.75758240006543250594321739230, −1.51248060644770097413333030884, 0.45399485558952875255433147353, 1.87475474144892579075050821668, 3.82987751539578501126669384364, 4.67427720264635630237961237896, 5.59639435561431816111519968792, 6.25450679413658727554513024868, 7.36475971328031899441288911714, 8.234852198000245921459785742194, 8.601042400676747695375215723829, 9.693675261751135000736662068278

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