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Properties

Label 2-1440-45.14-c0-0-2
Degree $2$
Conductor $1440$
Sign $0.642 + 0.766i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
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  + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)5-s + (1.67 − 0.965i)7-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)15-s + (−1.36 − 1.36i)21-s + (−0.258 + 0.448i)23-s + (0.499 + 0.866i)25-s + (0.707 + 0.707i)27-s + (−1.5 + 0.866i)29-s + 1.93·35-s + (0.866 + 0.5i)41-s + (−1.22 + 0.707i)43-s − 45-s + (−0.965 − 1.67i)47-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)5-s + (1.67 − 0.965i)7-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)15-s + (−1.36 − 1.36i)21-s + (−0.258 + 0.448i)23-s + (0.499 + 0.866i)25-s + (0.707 + 0.707i)27-s + (−1.5 + 0.866i)29-s + 1.93·35-s + (0.866 + 0.5i)41-s + (−1.22 + 0.707i)43-s − 45-s + (−0.965 − 1.67i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324534222\)
\(L(\frac12)\) \(\approx\) \(1.324534222\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
good7 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.73iT - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.688427002007089333885626214029, −8.605912715807615447642674140764, −7.80946129109454345762452577932, −7.25578609310451640136417267234, −6.46874068511562255923846163148, −5.47112597487993143571775924379, −4.84343208215919759162490210942, −3.45663528421374328759718544996, −2.02791327080258271498199240111, −1.42480268940059944557533779177, 1.66540134627859796854188377713, 2.65421908336412069359542945427, 4.20161972083372861273093438185, 4.88154310023735498560337614810, 5.59356269824348408626749112142, 6.11642580712567300303923982635, 7.66013612367254048323206603963, 8.520783619045291450820696527877, 9.024240145352123593012057624207, 9.751703303180239346710706974917

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