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Properties

Label 2-1080-40.29-c1-0-24
Degree $2$
Conductor $1080$
Sign $-0.842 - 0.538i$
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  + (0.986 + 1.01i)2-s + (−0.0536 + 1.99i)4-s + (0.569 + 2.16i)5-s − 4.64i·7-s + (−2.07 + 1.91i)8-s + (−1.62 + 2.71i)10-s + 3.44i·11-s − 2.72·13-s + (4.70 − 4.58i)14-s + (−3.99 − 0.214i)16-s + 2.43i·17-s + 7.45i·19-s + (−4.35 + 1.02i)20-s + (−3.49 + 3.40i)22-s + 6.60i·23-s + ⋯
L(s)  = 1  + (0.697 + 0.716i)2-s + (−0.0268 + 0.999i)4-s + (0.254 + 0.967i)5-s − 1.75i·7-s + (−0.734 + 0.678i)8-s + (−0.515 + 0.857i)10-s + 1.04i·11-s − 0.756·13-s + (1.25 − 1.22i)14-s + (−0.998 − 0.0536i)16-s + 0.589i·17-s + 1.71i·19-s + (−0.973 + 0.228i)20-s + (−0.745 + 0.725i)22-s + 1.37i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.842 - 0.538i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.964764437\)
\(L(\frac12)\) \(\approx\) \(1.964764437\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.986 - 1.01i)T \)
3 \( 1 \)
5 \( 1 + (-0.569 - 2.16i)T \)
good7 \( 1 + 4.64iT - 7T^{2} \)
11 \( 1 - 3.44iT - 11T^{2} \)
13 \( 1 + 2.72T + 13T^{2} \)
17 \( 1 - 2.43iT - 17T^{2} \)
19 \( 1 - 7.45iT - 19T^{2} \)
23 \( 1 - 6.60iT - 23T^{2} \)
29 \( 1 - 0.952iT - 29T^{2} \)
31 \( 1 - 5.77T + 31T^{2} \)
37 \( 1 + 0.0145T + 37T^{2} \)
41 \( 1 + 4.42T + 41T^{2} \)
43 \( 1 - 6.83T + 43T^{2} \)
47 \( 1 + 6.05iT - 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 + 9.70iT - 59T^{2} \)
61 \( 1 + 1.57iT - 61T^{2} \)
67 \( 1 + 7.29T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 3.38iT - 73T^{2} \)
79 \( 1 - 5.03T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 0.535T + 89T^{2} \)
97 \( 1 + 9.22iT - 97T^{2} \)
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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

−10.13995991557257570672743158106, −9.665319228395225908899290567537, −8.031641621719695150369037312552, −7.50294083746188614974185496306, −6.93857195397476555893406225616, −6.14772598763904292472765768697, −5.03137720381482574621141447806, −3.99824514251666719118562227499, −3.44325897926375727427385580984, −1.93540297860211649805978723334, 0.67372367963921988384205420468, 2.36892780386866881834883803334, 2.82823069831562384026982712034, 4.49079869452696990202002049359, 5.11073630575253756317192518339, 5.81934245120763706174793328903, 6.63104001085756141233163070395, 8.239725461913265919165067738692, 9.067231051939804998554040323778, 9.304169329193810710300243473268

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