Properties

Label 2-1080-40.29-c1-0-78
Degree $2$
Conductor $1080$
Sign $-0.447 + 0.894i$
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  + (−1.37 + 0.315i)2-s + (1.80 − 0.869i)4-s + (2.03 − 0.937i)5-s − 3.36i·7-s + (−2.20 + 1.76i)8-s + (−2.50 + 1.93i)10-s + 4.10i·11-s − 6.13·13-s + (1.06 + 4.64i)14-s + (2.48 − 3.13i)16-s − 7.47i·17-s − 6.20i·19-s + (2.84 − 3.45i)20-s + (−1.29 − 5.65i)22-s + 3.15i·23-s + ⋯
L(s)  = 1  + (−0.974 + 0.223i)2-s + (0.900 − 0.434i)4-s + (0.907 − 0.419i)5-s − 1.27i·7-s + (−0.780 + 0.624i)8-s + (−0.791 + 0.611i)10-s + 1.23i·11-s − 1.70·13-s + (0.284 + 1.24i)14-s + (0.621 − 0.783i)16-s − 1.81i·17-s − 1.42i·19-s + (0.635 − 0.772i)20-s + (−0.275 − 1.20i)22-s + 0.658i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7861871082\)
\(L(\frac12)\) \(\approx\) \(0.7861871082\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.315i)T \)
3 \( 1 \)
5 \( 1 + (-2.03 + 0.937i)T \)
good7 \( 1 + 3.36iT - 7T^{2} \)
11 \( 1 - 4.10iT - 11T^{2} \)
13 \( 1 + 6.13T + 13T^{2} \)
17 \( 1 + 7.47iT - 17T^{2} \)
19 \( 1 + 6.20iT - 19T^{2} \)
23 \( 1 - 3.15iT - 23T^{2} \)
29 \( 1 - 1.34iT - 29T^{2} \)
31 \( 1 - 0.229T + 31T^{2} \)
37 \( 1 + 5.64T + 37T^{2} \)
41 \( 1 + 4.52T + 41T^{2} \)
43 \( 1 + 2.02T + 43T^{2} \)
47 \( 1 - 1.67iT - 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 3.07iT - 59T^{2} \)
61 \( 1 - 0.762iT - 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + 1.26T + 71T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 - 0.245T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 8.64T + 89T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−9.634666987778748129668704786118, −9.140960495643514715729274542744, −7.77168503505320120136770075899, −7.07283021130213500774675274086, −6.79066114470031087846399789159, −5.08378766877848629291982006313, −4.82584780971398424556951058337, −2.84741732523122049098007050832, −1.85196895765560656467305623581, −0.44566487032617514657451484339, 1.74999183729992313662180465419, 2.52240069696756367920790502835, 3.52828752203770894884749266510, 5.38265308512910342466945193774, 6.03708593269796155040085668422, 6.72652457887713620035900469248, 8.035438049676102683112676431668, 8.509454233219547733569430773684, 9.379147686527980873535828680955, 10.12804843355330736150455973755

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