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Properties

Label 2-2160-9.4-c1-0-7
Degree $2$
Conductor $2160$
Sign $0.894 - 0.446i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
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  + (0.5 − 0.866i)5-s + (−0.596 − 1.03i)7-s + (1.66 + 2.87i)11-s + (−0.853 + 1.47i)13-s + 6.34·17-s − 1.32·19-s + (−3.43 + 5.94i)23-s + (−0.499 − 0.866i)25-s + (1.01 + 1.75i)29-s + (−1.33 + 2.32i)31-s − 1.19·35-s + 3.32·37-s + (1.16 − 2.00i)41-s + (−3.17 − 5.49i)43-s + (6.38 + 11.0i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.225 − 0.390i)7-s + (0.500 + 0.867i)11-s + (−0.236 + 0.410i)13-s + 1.53·17-s − 0.303·19-s + (−0.715 + 1.23i)23-s + (−0.0999 − 0.173i)25-s + (0.188 + 0.326i)29-s + (−0.240 + 0.416i)31-s − 0.201·35-s + 0.545·37-s + (0.181 − 0.313i)41-s + (−0.484 − 0.838i)43-s + (0.931 + 1.61i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.894 - 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.804106198\)
\(L(\frac12)\) \(\approx\) \(1.804106198\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (0.596 + 1.03i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.66 - 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.853 - 1.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.34T + 17T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + (3.43 - 5.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.01 - 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.33 - 2.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 + (-1.16 + 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.17 + 5.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.38 - 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.02T + 53T^{2} \)
59 \( 1 + (-5.83 + 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.86 - 8.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.28 - 9.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.06T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + (-0.707 - 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.91 - 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 + (8.12 + 14.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.320843709612336085956147481328, −8.343570058849893374614580990139, −7.49583828929467784791213665325, −6.93205835666550962214286167965, −5.90826058924332888787569673674, −5.18861596850756816903715249126, −4.20019809706648510434635423272, −3.47955835569987155601008952572, −2.12873781012565151515449908239, −1.11068714305320948140990563365, 0.75158739770917954857393052295, 2.24470797477688030304503988336, 3.14731909944910594752448432653, 3.97919805202957668542783210547, 5.17034359175742150023689524102, 6.00242661752902940476086637384, 6.44833052970008064469814507689, 7.58735438516967775393006688362, 8.211074092540366874411247155727, 9.021448909766739809031774972743

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