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Properties

Label 2-12e3-36.7-c2-0-14
Degree $2$
Conductor $1728$
Sign $0.864 - 0.503i$
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  + (1.81 + 3.13i)5-s + (1.59 + 0.920i)7-s + (−10.0 − 5.80i)11-s + (−6.43 − 11.1i)13-s − 12.6·17-s + 25.6i·19-s + (25.9 − 14.9i)23-s + (5.93 − 10.2i)25-s + (−10.8 + 18.7i)29-s + (52.4 − 30.2i)31-s + 6.67i·35-s + 25.7·37-s + (33.3 + 57.7i)41-s + (14.2 + 8.22i)43-s + (66.1 + 38.1i)47-s + ⋯
L(s)  = 1  + (0.362 + 0.627i)5-s + (0.227 + 0.131i)7-s + (−0.914 − 0.528i)11-s + (−0.495 − 0.857i)13-s − 0.742·17-s + 1.35i·19-s + (1.12 − 0.650i)23-s + (0.237 − 0.411i)25-s + (−0.372 + 0.645i)29-s + (1.69 − 0.975i)31-s + 0.190i·35-s + 0.695·37-s + (0.813 + 1.40i)41-s + (0.331 + 0.191i)43-s + (1.40 + 0.812i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.864 - 0.503i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.864 - 0.503i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.964619409\)
\(L(\frac12)\) \(\approx\) \(1.964619409\)
\(L(2)\) 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 \( 1 + (-1.81 - 3.13i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.59 - 0.920i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.0 + 5.80i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (6.43 + 11.1i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 12.6T + 289T^{2} \)
19 \( 1 - 25.6iT - 361T^{2} \)
23 \( 1 + (-25.9 + 14.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (10.8 - 18.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-52.4 + 30.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 25.7T + 1.36e3T^{2} \)
41 \( 1 + (-33.3 - 57.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-14.2 - 8.22i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-66.1 - 38.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 14.2T + 2.80e3T^{2} \)
59 \( 1 + (50.3 - 29.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-9.43 + 16.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 + 11.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 46.4iT - 5.04e3T^{2} \)
73 \( 1 - 49.3T + 5.32e3T^{2} \)
79 \( 1 + (-52.4 - 30.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-86.2 - 49.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 154.T + 7.92e3T^{2} \)
97 \( 1 + (21.1 - 36.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.276558115388382841433061073790, −8.164971762144350989467182222663, −7.85128006409679363644485052101, −6.70319997482219158576656944408, −6.00874547186640250443072176475, −5.20284358969222445400552341614, −4.26051421360793003246482822513, −2.90915787475936807755259834405, −2.47268499282903299403602960536, −0.850505606236439137957866764355, 0.69506244724675054245724537603, 2.00841783542793063220924733892, 2.85534269939927656665684892853, 4.40264473817614672931737178364, 4.83528032629709687021016731059, 5.67649303970706115441867388754, 6.89106192987949341295825131612, 7.32905960950875839461443406851, 8.404250319455202542236882434438, 9.183809234286829859120903548480

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