Properties

Label 2-12e3-36.31-c2-0-43
Degree $2$
Conductor $1728$
Sign $-0.819 + 0.573i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
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  + (1.10 − 1.90i)5-s + (7.23 − 4.17i)7-s + (−4.54 + 2.62i)11-s + (7.37 − 12.7i)13-s − 28.2·17-s − 19.1i·19-s + (−3.16 − 1.82i)23-s + (10.0 + 17.4i)25-s + (−12.3 − 21.3i)29-s + (−32.9 − 19.0i)31-s − 18.4i·35-s + 4.21·37-s + (9.92 − 17.1i)41-s + (−20.1 + 11.6i)43-s + (−25.8 + 14.9i)47-s + ⋯
L(s)  = 1  + (0.220 − 0.381i)5-s + (1.03 − 0.597i)7-s + (−0.413 + 0.238i)11-s + (0.567 − 0.982i)13-s − 1.66·17-s − 1.00i·19-s + (−0.137 − 0.0794i)23-s + (0.403 + 0.698i)25-s + (−0.425 − 0.736i)29-s + (−1.06 − 0.613i)31-s − 0.525i·35-s + 0.114·37-s + (0.242 − 0.419i)41-s + (−0.469 + 0.271i)43-s + (−0.550 + 0.317i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.819 + 0.573i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.312171132\)
\(L(\frac12)\) \(\approx\) \(1.312171132\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-1.10 + 1.90i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-7.23 + 4.17i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.54 - 2.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.37 + 12.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 28.2T + 289T^{2} \)
19 \( 1 + 19.1iT - 361T^{2} \)
23 \( 1 + (3.16 + 1.82i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (12.3 + 21.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (32.9 + 19.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 4.21T + 1.36e3T^{2} \)
41 \( 1 + (-9.92 + 17.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (20.1 - 11.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.8 - 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 32.1T + 2.80e3T^{2} \)
59 \( 1 + (7.96 + 4.59i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-40.8 - 70.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (6.86 + 3.96i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 62.9iT - 5.04e3T^{2} \)
73 \( 1 - 33.3T + 5.32e3T^{2} \)
79 \( 1 + (-53.7 + 31.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (103. - 59.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 107.T + 7.92e3T^{2} \)
97 \( 1 + (-1.78 - 3.09i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−8.812364927130981604294079659617, −8.003103945482350111273462322811, −7.37648794487375610793058574628, −6.42763716535038378779769842590, −5.38470733255367014915493352468, −4.74143443696596552077948659393, −3.91749928102238748361771900103, −2.59025050239082752784451122254, −1.55593261791739602058338777015, −0.32373791750791630967821300347, 1.62328829075793675299727610206, 2.28692588909776459491166776351, 3.57982221341383297581623359493, 4.57804800048529570697476974651, 5.36081619684099734703032153420, 6.31125692414059089676010312650, 6.95355831771233475339069885587, 8.059463981721635338591746666066, 8.634759574300042719259186266122, 9.278455537658989816975894686301

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