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Properties

Label 2-1080-5.4-c3-0-51
Degree $2$
Conductor $1080$
Sign $-0.160 + 0.987i$
Analytic cond. $63.7220$
Root an. cond. $7.98261$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 + 11.0i)5-s + 15.9i·7-s − 11.4·11-s − 37.5i·13-s − 75.9i·17-s − 11.0·19-s + 190. i·23-s + (−118. − 39.5i)25-s − 175.·29-s − 97.8·31-s + (−176. − 28.6i)35-s + 303. i·37-s − 171.·41-s − 521. i·43-s − 53.4i·47-s + ⋯
L(s)  = 1  + (−0.160 + 0.987i)5-s + 0.861i·7-s − 0.314·11-s − 0.800i·13-s − 1.08i·17-s − 0.133·19-s + 1.72i·23-s + (−0.948 − 0.316i)25-s − 1.12·29-s − 0.566·31-s + (−0.850 − 0.138i)35-s + 1.34i·37-s − 0.651·41-s − 1.84i·43-s − 0.165i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.160 + 0.987i$
Analytic conductor: \(63.7220\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :3/2),\ -0.160 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4517785496\)
\(L(\frac12)\) \(\approx\) \(0.4517785496\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.79 - 11.0i)T \)
good7 \( 1 - 15.9iT - 343T^{2} \)
11 \( 1 + 11.4T + 1.33e3T^{2} \)
13 \( 1 + 37.5iT - 2.19e3T^{2} \)
17 \( 1 + 75.9iT - 4.91e3T^{2} \)
19 \( 1 + 11.0T + 6.85e3T^{2} \)
23 \( 1 - 190. iT - 1.21e4T^{2} \)
29 \( 1 + 175.T + 2.43e4T^{2} \)
31 \( 1 + 97.8T + 2.97e4T^{2} \)
37 \( 1 - 303. iT - 5.06e4T^{2} \)
41 \( 1 + 171.T + 6.89e4T^{2} \)
43 \( 1 + 521. iT - 7.95e4T^{2} \)
47 \( 1 + 53.4iT - 1.03e5T^{2} \)
53 \( 1 + 102. iT - 1.48e5T^{2} \)
59 \( 1 - 396.T + 2.05e5T^{2} \)
61 \( 1 - 285.T + 2.26e5T^{2} \)
67 \( 1 + 706. iT - 3.00e5T^{2} \)
71 \( 1 - 215.T + 3.57e5T^{2} \)
73 \( 1 + 401. iT - 3.89e5T^{2} \)
79 \( 1 + 189.T + 4.93e5T^{2} \)
83 \( 1 - 490. iT - 5.71e5T^{2} \)
89 \( 1 + 154.T + 7.04e5T^{2} \)
97 \( 1 + 1.73e3iT - 9.12e5T^{2} \)
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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

−9.390965560762636628533650942173, −8.389872893667904646246607007293, −7.51802682701697024402438055137, −6.89423014682782502340822095723, −5.70265507224715462056561187347, −5.23491884539754508859713962784, −3.67924031431655801080594056012, −2.93868601322296811269742589477, −1.93387260744622946419795679456, −0.11748090006541653632236455491, 1.07476930455432440701311065066, 2.20042523752755530585703585324, 3.85815507252337976782230701323, 4.33020041224313825427774981433, 5.37495956382913789414590920147, 6.37190518991537636237461189930, 7.30572197255256434455133036316, 8.157399788543432877479220575182, 8.836477918312386136783464749333, 9.671793342298723072747369433827

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