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Properties

Label 2-2160-540.439-c0-0-0
Degree $2$
Conductor $2160$
Sign $0.973 + 0.230i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.173i)3-s + (−0.173 − 0.984i)5-s + (1.50 + 1.26i)7-s + (0.939 − 0.342i)9-s + (−0.342 − 0.939i)15-s + (1.70 + 0.984i)21-s + (−0.984 + 0.826i)23-s + (−0.939 + 0.342i)25-s + (0.866 − 0.5i)27-s + (−1.43 + 0.524i)29-s + (0.984 − 1.70i)35-s + (−1.76 − 0.642i)41-s + (0.300 − 1.70i)43-s + (−0.5 − 0.866i)45-s + (0.524 + 0.439i)47-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)3-s + (−0.173 − 0.984i)5-s + (1.50 + 1.26i)7-s + (0.939 − 0.342i)9-s + (−0.342 − 0.939i)15-s + (1.70 + 0.984i)21-s + (−0.984 + 0.826i)23-s + (−0.939 + 0.342i)25-s + (0.866 − 0.5i)27-s + (−1.43 + 0.524i)29-s + (0.984 − 1.70i)35-s + (−1.76 − 0.642i)41-s + (0.300 − 1.70i)43-s + (−0.5 − 0.866i)45-s + (0.524 + 0.439i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.973 + 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.856231251\)
\(L(\frac12)\) \(\approx\) \(1.856231251\)
\(L(1)\) 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 \( 1 + (-0.984 + 0.173i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
good7 \( 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.939 + 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.984 - 0.826i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.300 + 1.70i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.524 - 0.439i)T + (0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.642 + 0.233i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.642 + 0.233i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−8.919312448132358955244238858475, −8.596338449406431936674533228395, −7.86209927908775312195393958997, −7.30464057499631089977077570141, −5.82679839033468978240025572648, −5.23212501918744459044346082713, −4.39113232433769566819015496986, −3.46861484365125401696253494599, −2.06263651254735115077061233564, −1.64703620757712921883788924269, 1.55673011068972144444503748231, 2.46970913219637907676172393712, 3.67328716630757076524547396703, 4.17869894856776164074454517712, 5.05148698385430991770761135607, 6.41741466851658060742524649901, 7.21482446571206193942646028289, 7.910420625613740601173213510041, 8.144064325691346083518201533574, 9.307357133177119909855955011419

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