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Properties

Label 2-12e3-36.31-c2-0-5
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: $\chi_{1728} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.864 - 0.503i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2857637551\)
\(L(\frac12)\) \(\approx\) \(0.2857637551\)
\(L(2)\) 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

−9.419022084225547446704049663660, −8.819799971186565092162048649144, −7.985088163318424740337324865012, −7.02667517351120568209375397807, −6.11466294971616754279669441374, −5.60201492893585024721879959610, −4.33036423182998276516079475069, −3.82079510540780154776463489394, −2.32922093327459645348042896177, −1.44277960640936814286221239906, 0.07065981337002726920558171651, 1.67946277772801176477727514334, 2.71081495618094996341816808665, 3.64639984542430179926991645596, 4.68584577586785101269940803678, 5.57200111594562553521325636023, 6.66952712993619766010669932423, 6.95229461258065318034929981166, 7.963953369685178513934843640147, 8.948845121307274329155480319919

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