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Properties

Label 2-1080-120.59-c1-0-18
Degree $2$
Conductor $1080$
Sign $0.626 - 0.779i$
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  + (−1.34 − 0.425i)2-s + (1.63 + 1.14i)4-s + (−2.17 + 0.529i)5-s − 1.59·7-s + (−1.72 − 2.24i)8-s + (3.15 + 0.210i)10-s − 3.61i·11-s − 2.18·13-s + (2.15 + 0.678i)14-s + (1.36 + 3.75i)16-s − 2.27·17-s + 5.81·19-s + (−4.16 − 1.62i)20-s + (−1.53 + 4.87i)22-s − 1.07i·23-s + ⋯
L(s)  = 1  + (−0.953 − 0.300i)2-s + (0.819 + 0.573i)4-s + (−0.971 + 0.236i)5-s − 0.602·7-s + (−0.608 − 0.793i)8-s + (0.997 + 0.0665i)10-s − 1.08i·11-s − 0.606·13-s + (0.574 + 0.181i)14-s + (0.341 + 0.939i)16-s − 0.552·17-s + 1.33·19-s + (−0.931 − 0.363i)20-s + (−0.327 + 1.03i)22-s − 0.225i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.626 - 0.779i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5486912507\)
\(L(\frac12)\) \(\approx\) \(0.5486912507\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.425i)T \)
3 \( 1 \)
5 \( 1 + (2.17 - 0.529i)T \)
good7 \( 1 + 1.59T + 7T^{2} \)
11 \( 1 + 3.61iT - 11T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 - 5.81T + 19T^{2} \)
23 \( 1 + 1.07iT - 23T^{2} \)
29 \( 1 - 2.55T + 29T^{2} \)
31 \( 1 - 5.93iT - 31T^{2} \)
37 \( 1 - 2.49T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 0.983iT - 43T^{2} \)
47 \( 1 - 10.6iT - 47T^{2} \)
53 \( 1 - 8.26iT - 53T^{2} \)
59 \( 1 - 2.81iT - 59T^{2} \)
61 \( 1 + 8.34iT - 61T^{2} \)
67 \( 1 - 6.00iT - 67T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 + 5.96iT - 73T^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 - 6.40T + 83T^{2} \)
89 \( 1 + 6.53iT - 89T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−9.914708304872586578526984186926, −9.204134102279036388585980889040, −8.340535263375399094579024897443, −7.68640063858153970058686203203, −6.86459456810180173451301594430, −6.08401372248974579005323646601, −4.61710171424450319412824907727, −3.31948542548641626634357838443, −2.85252698401509815659002721856, −0.952022418936620586613442354069, 0.43309403266710820596458571926, 2.10077660556859040165199846522, 3.35696931438480585944924469439, 4.61317825942249697059546768294, 5.57441536759487505030064142031, 6.85781951556145242665680417044, 7.28032053875206086927078021727, 8.046996979438909667380806767135, 8.993171093381795708175124366572, 9.681865929990101787338162501128

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