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Properties

Label 2-1080-120.59-c1-0-20
Degree $2$
Conductor $1080$
Sign $-0.999 - 0.0257i$
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  + (−0.482 + 1.32i)2-s + (−1.53 − 1.28i)4-s + (−1.17 + 1.90i)5-s + 0.450·7-s + (2.44 − 1.42i)8-s + (−1.96 − 2.47i)10-s + 5.11i·11-s + 1.15·13-s + (−0.217 + 0.598i)14-s + (0.710 + 3.93i)16-s + 6.13·17-s + 2.16·19-s + (4.24 − 1.41i)20-s + (−6.79 − 2.46i)22-s + 0.211i·23-s + ⋯
L(s)  = 1  + (−0.341 + 0.940i)2-s + (−0.767 − 0.641i)4-s + (−0.524 + 0.851i)5-s + 0.170·7-s + (0.864 − 0.502i)8-s + (−0.621 − 0.783i)10-s + 1.54i·11-s + 0.320·13-s + (−0.0580 + 0.160i)14-s + (0.177 + 0.984i)16-s + 1.48·17-s + 0.497·19-s + (0.948 − 0.316i)20-s + (−1.44 − 0.525i)22-s + 0.0440i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.999 - 0.0257i$
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: \((2,\ 1080,\ (\ :1/2),\ -0.999 - 0.0257i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8979676525\)
\(L(\frac12)\) \(\approx\) \(0.8979676525\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.482 - 1.32i)T \)
3 \( 1 \)
5 \( 1 + (1.17 - 1.90i)T \)
good7 \( 1 - 0.450T + 7T^{2} \)
11 \( 1 - 5.11iT - 11T^{2} \)
13 \( 1 - 1.15T + 13T^{2} \)
17 \( 1 - 6.13T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 - 0.211iT - 23T^{2} \)
29 \( 1 + 6.08T + 29T^{2} \)
31 \( 1 - 0.417iT - 31T^{2} \)
37 \( 1 + 9.60T + 37T^{2} \)
41 \( 1 - 9.71iT - 41T^{2} \)
43 \( 1 - 3.06iT - 43T^{2} \)
47 \( 1 + 2.65iT - 47T^{2} \)
53 \( 1 - 8.81iT - 53T^{2} \)
59 \( 1 + 4.73iT - 59T^{2} \)
61 \( 1 - 7.64iT - 61T^{2} \)
67 \( 1 - 6.49iT - 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 - 5.22T + 83T^{2} \)
89 \( 1 + 1.39iT - 89T^{2} \)
97 \( 1 + 4.41iT - 97T^{2} \)
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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

−10.02383179112013773521141161100, −9.575169876210381348188894389454, −8.403275228618879167566182737867, −7.52655992447098437941249015790, −7.24356732269610241096313341830, −6.23441500185474853641765077465, −5.27223997978936802334350347942, −4.32138329137838926057230979917, −3.27050394472129972314298407968, −1.57894468571954935754095951154, 0.49105987949227285565491363988, 1.60052841214889812484859742639, 3.31536379450221860735565705876, 3.73949149611804434274276895207, 5.09504793653010057046663096354, 5.70394184408622107687656997508, 7.35870734041093786432392764927, 8.120344559545480252364052039319, 8.705764650498770600071568981486, 9.403279240672311117082057943185

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