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Properties

Label 2-1080-120.59-c1-0-38
Degree $2$
Conductor $1080$
Sign $-0.456 - 0.889i$
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  + (1.09 + 0.891i)2-s + (0.411 + 1.95i)4-s + (−1.30 + 1.81i)5-s + 4.43·7-s + (−1.29 + 2.51i)8-s + (−3.05 + 0.833i)10-s + 0.767i·11-s + 3.93·13-s + (4.87 + 3.95i)14-s + (−3.66 + 1.61i)16-s − 2.35·17-s + 1.58·19-s + (−4.09 − 1.80i)20-s + (−0.684 + 0.843i)22-s + 0.273i·23-s + ⋯
L(s)  = 1  + (0.776 + 0.630i)2-s + (0.205 + 0.978i)4-s + (−0.582 + 0.812i)5-s + 1.67·7-s + (−0.457 + 0.889i)8-s + (−0.964 + 0.263i)10-s + 0.231i·11-s + 1.09·13-s + (1.30 + 1.05i)14-s + (−0.915 + 0.402i)16-s − 0.571·17-s + 0.362·19-s + (−0.915 − 0.403i)20-s + (−0.145 + 0.179i)22-s + 0.0570i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.456 - 0.889i$
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: \((2,\ 1080,\ (\ :1/2),\ -0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.658108265\)
\(L(\frac12)\) \(\approx\) \(2.658108265\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 - 0.891i)T \)
3 \( 1 \)
5 \( 1 + (1.30 - 1.81i)T \)
good7 \( 1 - 4.43T + 7T^{2} \)
11 \( 1 - 0.767iT - 11T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 + 2.35T + 17T^{2} \)
19 \( 1 - 1.58T + 19T^{2} \)
23 \( 1 - 0.273iT - 23T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 - 5.26iT - 31T^{2} \)
37 \( 1 + 9.37T + 37T^{2} \)
41 \( 1 + 8.35iT - 41T^{2} \)
43 \( 1 - 7.54iT - 43T^{2} \)
47 \( 1 + 9.77iT - 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 - 6.06iT - 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 + 15.3iT - 67T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 - 0.340iT - 79T^{2} \)
83 \( 1 + 3.82T + 83T^{2} \)
89 \( 1 - 0.941iT - 89T^{2} \)
97 \( 1 - 2.54iT - 97T^{2} \)
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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

−10.62292693398856288908792185704, −8.848180854371758532471210885700, −8.333581562627158028849631754009, −7.50086640527352569408399043453, −6.89315006941656710826160369450, −5.86479959910786713821122567472, −4.90703132161212141226442709342, −4.14194313428590254370141751714, −3.18693053220561251409616233140, −1.86025705961819428318556379888, 1.01617850037541093387244229309, 1.97111577658166709007865429568, 3.52548853508972415553432836686, 4.38581759132069255177706987313, 5.03868177959855161015847473493, 5.84042104595769243185925931896, 7.06302612310829322620679238525, 8.213543799416058852266534228771, 8.628021817997516660293004755976, 9.703379761455071136710889514879

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