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Properties

Label 2-1080-120.59-c1-0-25
Degree $2$
Conductor $1080$
Sign $0.909 + 0.416i$
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  + (0.341 − 1.37i)2-s + (−1.76 − 0.936i)4-s + (−2.13 + 0.663i)5-s − 4.39·7-s + (−1.88 + 2.10i)8-s + (0.182 + 3.15i)10-s − 0.633i·11-s + 3.40·13-s + (−1.50 + 6.03i)14-s + (2.24 + 3.30i)16-s + 5.48·17-s + 0.323·19-s + (4.39 + 0.826i)20-s + (−0.868 − 0.215i)22-s + 6.26i·23-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (−0.883 − 0.468i)4-s + (−0.954 + 0.296i)5-s − 1.66·7-s + (−0.667 + 0.744i)8-s + (0.0577 + 0.998i)10-s − 0.190i·11-s + 0.944·13-s + (−0.400 + 1.61i)14-s + (0.561 + 0.827i)16-s + 1.33·17-s + 0.0741·19-s + (0.982 + 0.184i)20-s + (−0.185 − 0.0460i)22-s + 1.30i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.909 + 0.416i$
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: \((2,\ 1080,\ (\ :1/2),\ 0.909 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9491851185\)
\(L(\frac12)\) \(\approx\) \(0.9491851185\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.341 + 1.37i)T \)
3 \( 1 \)
5 \( 1 + (2.13 - 0.663i)T \)
good7 \( 1 + 4.39T + 7T^{2} \)
11 \( 1 + 0.633iT - 11T^{2} \)
13 \( 1 - 3.40T + 13T^{2} \)
17 \( 1 - 5.48T + 17T^{2} \)
19 \( 1 - 0.323T + 19T^{2} \)
23 \( 1 - 6.26iT - 23T^{2} \)
29 \( 1 + 3.77T + 29T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 - 5.90T + 37T^{2} \)
41 \( 1 - 0.877iT - 41T^{2} \)
43 \( 1 + 2.31iT - 43T^{2} \)
47 \( 1 - 8.67iT - 47T^{2} \)
53 \( 1 - 7.09iT - 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 + 14.0iT - 61T^{2} \)
67 \( 1 - 5.99iT - 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 13.9iT - 73T^{2} \)
79 \( 1 - 7.33iT - 79T^{2} \)
83 \( 1 - 1.90T + 83T^{2} \)
89 \( 1 - 8.16iT - 89T^{2} \)
97 \( 1 - 6.27iT - 97T^{2} \)
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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

−9.719484041963713582201631261562, −9.437230740752919073914291826485, −8.257488071267231026492678402987, −7.42832283614836468464658436683, −6.21797889690276103515304751959, −5.58600667555304791784550713748, −4.02981579888326704039573519455, −3.54464464526459726732538194492, −2.78166102451540342136072603214, −0.910349116929885882982695225347, 0.56290854760397584559242381846, 3.23697846252527586587817366704, 3.66228637034744671408337423999, 4.78604990135363906417773693246, 5.85372456710036794041170377903, 6.59983133037852612681119090612, 7.31803636684685816273094397183, 8.232890453505634931136869362186, 8.903241040085997437763029188855, 9.730042167556317964477519734200

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