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Properties

Label 2-1080-40.29-c1-0-27
Degree $2$
Conductor $1080$
Sign $0.675 - 0.737i$
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  + (1.05 − 0.943i)2-s + (0.219 − 1.98i)4-s + (0.463 + 2.18i)5-s + 3.14i·7-s + (−1.64 − 2.30i)8-s + (2.55 + 1.86i)10-s + 2.42i·11-s − 3.12·13-s + (2.97 + 3.31i)14-s + (−3.90 − 0.874i)16-s + 7.15i·17-s + 2.34i·19-s + (4.45 − 0.440i)20-s + (2.28 + 2.55i)22-s − 1.28i·23-s + ⋯
L(s)  = 1  + (0.744 − 0.667i)2-s + (0.109 − 0.993i)4-s + (0.207 + 0.978i)5-s + 1.19i·7-s + (−0.581 − 0.813i)8-s + (0.807 + 0.590i)10-s + 0.730i·11-s − 0.866·13-s + (0.794 + 0.886i)14-s + (−0.975 − 0.218i)16-s + 1.73i·17-s + 0.538i·19-s + (0.995 − 0.0984i)20-s + (0.487 + 0.543i)22-s − 0.268i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.675 - 0.737i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.055633777\)
\(L(\frac12)\) \(\approx\) \(2.055633777\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 + 0.943i)T \)
3 \( 1 \)
5 \( 1 + (-0.463 - 2.18i)T \)
good7 \( 1 - 3.14iT - 7T^{2} \)
11 \( 1 - 2.42iT - 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 - 7.15iT - 17T^{2} \)
19 \( 1 - 2.34iT - 19T^{2} \)
23 \( 1 + 1.28iT - 23T^{2} \)
29 \( 1 + 4.21iT - 29T^{2} \)
31 \( 1 - 3.05T + 31T^{2} \)
37 \( 1 - 1.37T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 9.70T + 43T^{2} \)
47 \( 1 + 0.627iT - 47T^{2} \)
53 \( 1 - 9.54T + 53T^{2} \)
59 \( 1 - 10.1iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 7.38T + 71T^{2} \)
73 \( 1 - 5.73iT - 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 3.12T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−10.08884950126372710332082770181, −9.553598758508647485983965966041, −8.424024803408939517238674433786, −7.31480524780922865940320079967, −6.24155164051732720640843998762, −5.82512840145457283707797173859, −4.68968364768109985577847596423, −3.67191768045718829400906347914, −2.53260471288438764290950403353, −1.95312052686502374880234564358, 0.69287796486885197479319011522, 2.62012899318291938467904497310, 3.78061463609151335157543179256, 4.78156448557442683524639664407, 5.20956328389881040835716969979, 6.39623371147761011823461120714, 7.28897369411270617573865916446, 7.83018803031642297351836007907, 8.901166115003248742733497640480, 9.530005011618250135357919352989

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