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Properties

Label 2-2160-80.19-c0-0-0
Degree $2$
Conductor $2160$
Sign $0.608 - 0.793i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)8-s + (0.500 + 0.866i)10-s + (0.500 + 0.866i)16-s + 1.93i·17-s + (−1.36 + 1.36i)19-s + (−0.258 − 0.965i)20-s + 0.517i·23-s + 1.00i·25-s − i·31-s + (−0.258 − 0.965i)32-s + (0.499 − 1.86i)34-s + (1.67 − 0.965i)38-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)8-s + (0.500 + 0.866i)10-s + (0.500 + 0.866i)16-s + 1.93i·17-s + (−1.36 + 1.36i)19-s + (−0.258 − 0.965i)20-s + 0.517i·23-s + 1.00i·25-s − i·31-s + (−0.258 − 0.965i)32-s + (0.499 − 1.86i)34-s + (1.67 − 0.965i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.608 - 0.793i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5073706961\)
\(L(\frac12)\) \(\approx\) \(0.5073706961\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - 1.93iT - T^{2} \)
19 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
23 \( 1 - 0.517iT - T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.173162764443802924345086097843, −8.583625704647858337024889065341, −8.016652949355654006477835670804, −7.41839880940624150507255602132, −6.27665574855962067544408979897, −5.67743619891521933673443141791, −4.05789509306190249838425002551, −3.83523755571541340468162986534, −2.28136923387180528574986292052, −1.28640449778082415239510948007, 0.51826993360904595937007089043, 2.35567183302053413406086109627, 2.96377883081000609096547569858, 4.32379546558310405755552285077, 5.26117657174370886429455157181, 6.40875171508268666782621023849, 7.06965871587482881501547023200, 7.41361849347263291077823645139, 8.567253422774888853841064442608, 8.910676713099752165168887931268

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