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Properties

Label 2-1080-9.7-c1-0-5
Degree $2$
Conductor $1080$
Sign $0.534 - 0.845i$
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  + (0.5 + 0.866i)5-s + (−1.51 + 2.63i)7-s + (2.63 − 4.56i)11-s + (0.256 + 0.444i)13-s − 2.80·17-s + 8.29·19-s + (2.51 + 4.36i)23-s + (−0.499 + 0.866i)25-s + (−2.39 + 4.14i)29-s + (4.29 + 7.43i)31-s − 3.03·35-s − 6.58·37-s + (3.99 + 6.91i)41-s + (0.598 − 1.03i)43-s + (−4.81 + 8.33i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.573 + 0.994i)7-s + (0.794 − 1.37i)11-s + (0.0712 + 0.123i)13-s − 0.679·17-s + 1.90·19-s + (0.525 + 0.909i)23-s + (−0.0999 + 0.173i)25-s + (−0.444 + 0.769i)29-s + (0.771 + 1.33i)31-s − 0.513·35-s − 1.08·37-s + (0.623 + 1.07i)41-s + (0.0912 − 0.158i)43-s + (−0.701 + 1.21i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.534 - 0.845i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.534 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.596529119\)
\(L(\frac12)\) \(\approx\) \(1.596529119\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (1.51 - 2.63i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.63 + 4.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.256 - 0.444i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.80T + 17T^{2} \)
19 \( 1 - 8.29T + 19T^{2} \)
23 \( 1 + (-2.51 - 4.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.39 - 4.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.29 - 7.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + (-3.99 - 6.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.598 + 1.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.81 - 8.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.467T + 53T^{2} \)
59 \( 1 + (0.378 + 0.655i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.135 + 0.234i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.63 - 6.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.48T + 71T^{2} \)
73 \( 1 - 5.31T + 73T^{2} \)
79 \( 1 + (-7.98 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.03 + 10.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + (-3.18 + 5.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.819453509349294077667756726191, −9.143703599468462339957572835677, −8.632047972397084248867652802601, −7.42532551704010516078424386175, −6.52249681075228034641331959308, −5.84825397654838406412019845911, −5.02220545896374828138578225136, −3.40635590676707286282630290763, −2.98480513845315864950342964793, −1.34894203617498501371268379662, 0.807043190675535263428212814719, 2.20460495403976045166222154837, 3.65098772228256461860620859824, 4.41445566440129939131419066212, 5.37211071301094871190215153615, 6.61314663872479932932612697688, 7.10802274940091374519986875758, 7.997405322469064842544073877799, 9.204902794087359667972631253409, 9.687439309129396968701641743923

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