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Properties

Label 2-2160-15.14-c2-0-33
Degree $2$
Conductor $2160$
Sign $0.967 - 0.253i$
Analytic cond. $58.8557$
Root an. cond. $7.67174$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 − 4.83i)5-s + 6.26i·7-s − 1.03i·11-s − 3i·13-s − 15.7·17-s − 18.7·19-s + 34.6·23-s + (−21.7 + 12.2i)25-s + 3.10i·29-s + 17.7·31-s + (30.2 − 7.93i)35-s + 43.3i·37-s − 46.2i·41-s + 20.7i·43-s + 78.8·47-s + ⋯
L(s)  = 1  + (−0.253 − 0.967i)5-s + 0.894i·7-s − 0.0939i·11-s − 0.230i·13-s − 0.928·17-s − 0.988·19-s + 1.50·23-s + (−0.871 + 0.490i)25-s + 0.106i·29-s + 0.573·31-s + (0.865 − 0.226i)35-s + 1.17i·37-s − 1.12i·41-s + 0.482i·43-s + 1.67·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(58.8557\)
Root analytic conductor: \(7.67174\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1),\ 0.967 - 0.253i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.609332088\)
\(L(\frac12)\) \(\approx\) \(1.609332088\)
\(L(2)\) 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 \( 1 + (1.26 + 4.83i)T \)
good7 \( 1 - 6.26iT - 49T^{2} \)
11 \( 1 + 1.03iT - 121T^{2} \)
13 \( 1 + 3iT - 169T^{2} \)
17 \( 1 + 15.7T + 289T^{2} \)
19 \( 1 + 18.7T + 361T^{2} \)
23 \( 1 - 34.6T + 529T^{2} \)
29 \( 1 - 3.10iT - 841T^{2} \)
31 \( 1 - 17.7T + 961T^{2} \)
37 \( 1 - 43.3iT - 1.36e3T^{2} \)
41 \( 1 + 46.2iT - 1.68e3T^{2} \)
43 \( 1 - 20.7iT - 1.84e3T^{2} \)
47 \( 1 - 78.8T + 2.20e3T^{2} \)
53 \( 1 + 44.2T + 2.80e3T^{2} \)
59 \( 1 - 90.5iT - 3.48e3T^{2} \)
61 \( 1 - 8.78T + 3.72e3T^{2} \)
67 \( 1 - 19.3iT - 4.48e3T^{2} \)
71 \( 1 + 56.6iT - 5.04e3T^{2} \)
73 \( 1 + 109. iT - 5.32e3T^{2} \)
79 \( 1 + 39T + 6.24e3T^{2} \)
83 \( 1 - 75.7T + 6.88e3T^{2} \)
89 \( 1 - 90.5iT - 7.92e3T^{2} \)
97 \( 1 + 8.88iT - 9.40e3T^{2} \)
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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

−8.855676224145136528296658449352, −8.421143869370432805675187648329, −7.45583456059917773907532619839, −6.50442963044850219516910809944, −5.69135479484967726971697374314, −4.88786862500270815746622905587, −4.22909758676298651061515978851, −2.99883000136302208247661395016, −2.02026456487708909315387636632, −0.76774650437747647889858190534, 0.56334671977084237358535064952, 2.05760295007501965512275015863, 3.00846635540368748784201598221, 4.01247667522379546447117761314, 4.59935823275189554004310515378, 5.86151415439831022292134878563, 6.82419301904261006438008456575, 7.05699063204036262417243883318, 8.005996719578144528257159481861, 8.843577666676201016524482412966

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