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Properties

Label 2-1080-45.4-c1-0-14
Degree $2$
Conductor $1080$
Sign $-0.205 + 0.978i$
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  + (−1.16 + 1.90i)5-s + (3.55 − 2.05i)7-s + (−3.04 − 5.27i)11-s + (−4.45 − 2.56i)13-s + 2.73i·17-s − 1.80·19-s + (−0.582 − 0.336i)23-s + (−2.29 − 4.44i)25-s + (−1.75 − 3.03i)29-s + (3.30 − 5.71i)31-s + (−0.213 + 9.17i)35-s + 4.44i·37-s + (2.08 − 3.61i)41-s + (−4.01 + 2.31i)43-s + (1.38 − 0.798i)47-s + ⋯
L(s)  = 1  + (−0.520 + 0.854i)5-s + (1.34 − 0.775i)7-s + (−0.918 − 1.59i)11-s + (−1.23 − 0.712i)13-s + 0.663i·17-s − 0.414·19-s + (−0.121 − 0.0701i)23-s + (−0.459 − 0.888i)25-s + (−0.325 − 0.563i)29-s + (0.592 − 1.02i)31-s + (−0.0361 + 1.55i)35-s + 0.730i·37-s + (0.326 − 0.565i)41-s + (−0.611 + 0.353i)43-s + (0.201 − 0.116i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.205 + 0.978i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.014622174\)
\(L(\frac12)\) \(\approx\) \(1.014622174\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 - 1.90i)T \)
good7 \( 1 + (-3.55 + 2.05i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.04 + 5.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.45 + 2.56i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.73iT - 17T^{2} \)
19 \( 1 + 1.80T + 19T^{2} \)
23 \( 1 + (0.582 + 0.336i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 + 3.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.30 + 5.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.44iT - 37T^{2} \)
41 \( 1 + (-2.08 + 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.01 - 2.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.38 + 0.798i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.02iT - 53T^{2} \)
59 \( 1 + (-2.65 + 4.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.82 - 5.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 6.50iT - 73T^{2} \)
79 \( 1 + (-1.49 - 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.71 + 2.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.19T + 89T^{2} \)
97 \( 1 + (3.33 - 1.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.996174601782774415826548488662, −8.378450055799005888902511530337, −8.010550841487700049279200671033, −7.40610991988519256689771008404, −6.26629523491727371928244124309, −5.30260173021029550982018912535, −4.34791886099923422368473022328, −3.32032539400495130866665297241, −2.26885716912376960562388791589, −0.43619267753967429582473231738, 1.68270987997616682933510482521, 2.54344877931497317444337291505, 4.42950833424924435925937121771, 4.80874519053312839346921898773, 5.46904794730549738306783201930, 7.13096847750504881139705650382, 7.59713861648168627823133559927, 8.470263617644834281886259410435, 9.196847529664635697622515583572, 9.993432754104631188513657136968

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