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Properties

Label 2-1080-120.59-c1-0-94
Degree $2$
Conductor $1080$
Sign $-0.769 + 0.638i$
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  + (1.41 − 0.0850i)2-s + (1.98 − 0.240i)4-s + (−1.09 − 1.94i)5-s − 4.08·7-s + (2.78 − 0.507i)8-s + (−1.71 − 2.65i)10-s − 4.79i·11-s − 1.84·13-s + (−5.77 + 0.347i)14-s + (3.88 − 0.953i)16-s − 4.59·17-s − 5.57·19-s + (−2.64 − 3.60i)20-s + (−0.408 − 6.77i)22-s + 6.84i·23-s + ⋯
L(s)  = 1  + (0.998 − 0.0601i)2-s + (0.992 − 0.120i)4-s + (−0.489 − 0.871i)5-s − 1.54·7-s + (0.983 − 0.179i)8-s + (−0.541 − 0.840i)10-s − 1.44i·11-s − 0.512·13-s + (−1.54 + 0.0929i)14-s + (0.971 − 0.238i)16-s − 1.11·17-s − 1.27·19-s + (−0.591 − 0.806i)20-s + (−0.0870 − 1.44i)22-s + 1.42i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.769 + 0.638i$
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: \((2,\ 1080,\ (\ :1/2),\ -0.769 + 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.480350585\)
\(L(\frac12)\) \(\approx\) \(1.480350585\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0850i)T \)
3 \( 1 \)
5 \( 1 + (1.09 + 1.94i)T \)
good7 \( 1 + 4.08T + 7T^{2} \)
11 \( 1 + 4.79iT - 11T^{2} \)
13 \( 1 + 1.84T + 13T^{2} \)
17 \( 1 + 4.59T + 17T^{2} \)
19 \( 1 + 5.57T + 19T^{2} \)
23 \( 1 - 6.84iT - 23T^{2} \)
29 \( 1 - 6.33T + 29T^{2} \)
31 \( 1 + 9.51iT - 31T^{2} \)
37 \( 1 - 2.44T + 37T^{2} \)
41 \( 1 - 2.10iT - 41T^{2} \)
43 \( 1 + 9.80iT - 43T^{2} \)
47 \( 1 + 8.06iT - 47T^{2} \)
53 \( 1 - 0.573iT - 53T^{2} \)
59 \( 1 + 5.20iT - 59T^{2} \)
61 \( 1 + 4.39iT - 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 8.42iT - 73T^{2} \)
79 \( 1 + 7.01iT - 79T^{2} \)
83 \( 1 - 5.58T + 83T^{2} \)
89 \( 1 - 4.57iT - 89T^{2} \)
97 \( 1 - 4.86iT - 97T^{2} \)
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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

−9.565394294068656807041971315904, −8.738746730617112886735156217435, −7.82328075938372238207880364652, −6.71317295246853851519311959786, −6.11443758897789163838423670805, −5.23731532265430786883637285385, −4.09864303391794038834602003398, −3.49110760322311522844417200619, −2.34272522516679067904396287811, −0.42824122756984059328214806092, 2.35913296880832267043934746078, 2.94200236617637212933605012393, 4.17148635926062604130175848116, 4.70510590880501755991500710430, 6.43798779102316943972454478912, 6.53501279743678085388265718601, 7.25444342687695013423834649641, 8.396783484450844597731029227167, 9.642889609862629051362650272621, 10.41379062394162972179373162884

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