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Properties

Label 2-2160-15.14-c0-0-0
Degree $2$
Conductor $2160$
Sign $-0.707 - 0.707i$
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  + (−0.707 + 0.707i)5-s + i·7-s + 1.41i·11-s − i·13-s − 19-s − 1.41·23-s − 1.00i·25-s + 1.41i·29-s + (−0.707 − 0.707i)35-s + i·37-s − 1.41i·41-s − 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (0.707 + 0.707i)65-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)5-s + i·7-s + 1.41i·11-s − i·13-s − 19-s − 1.41·23-s − 1.00i·25-s + 1.41i·29-s + (−0.707 − 0.707i)35-s + i·37-s − 1.41i·41-s − 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (0.707 + 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6903379110\)
\(L(\frac12)\) \(\approx\) \(0.6903379110\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - iT - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−9.619713358843898470652388011208, −8.626772884246181481728064390979, −8.033633710346540186562753662248, −7.24232143035549380544886447272, −6.51238460270033905146623495305, −5.63076908321124070471609486504, −4.71388620877052884621676361421, −3.81147327691642548807846911105, −2.79236654733543617383755313581, −1.95934878714665840016153863370, 0.46525003722821184951591825544, 1.86200313568443260690063170890, 3.39341794277616475597240913041, 4.12912090296278936926376274417, 4.67362916347042190253038986671, 5.99004745699615919943991739275, 6.50694258561531982445029001036, 7.77110295266367842309127854352, 8.002421206310358632771626564699, 8.944814900947348085536654916066

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