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Properties

Label 2-12e3-36.31-c2-0-36
Degree $2$
Conductor $1728$
Sign $0.224 + 0.974i$
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  + (3.07 − 5.32i)5-s + (0.511 − 0.295i)7-s + (15.1 − 8.72i)11-s + (0.892 − 1.54i)13-s + 16.9·17-s − 19.5i·19-s + (6.86 + 3.96i)23-s + (−6.39 − 11.0i)25-s + (3.17 + 5.49i)29-s + (27.6 + 15.9i)31-s − 3.63i·35-s − 58.2·37-s + (2.66 − 4.62i)41-s + (33.9 − 19.5i)43-s + (9.64 − 5.56i)47-s + ⋯
L(s)  = 1  + (0.614 − 1.06i)5-s + (0.0730 − 0.0421i)7-s + (1.37 − 0.793i)11-s + (0.0686 − 0.118i)13-s + 0.995·17-s − 1.02i·19-s + (0.298 + 0.172i)23-s + (−0.255 − 0.443i)25-s + (0.109 + 0.189i)29-s + (0.892 + 0.515i)31-s − 0.103i·35-s − 1.57·37-s + (0.0651 − 0.112i)41-s + (0.789 − 0.455i)43-s + (0.205 − 0.118i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.224 + 0.974i$
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: \((2,\ 1728,\ (\ :1),\ 0.224 + 0.974i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.680221356\)
\(L(\frac12)\) \(\approx\) \(2.680221356\)
\(L(2)\) 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 \( 1 + (-3.07 + 5.32i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-0.511 + 0.295i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-15.1 + 8.72i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.892 + 1.54i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 16.9T + 289T^{2} \)
19 \( 1 + 19.5iT - 361T^{2} \)
23 \( 1 + (-6.86 - 3.96i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-3.17 - 5.49i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-27.6 - 15.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 58.2T + 1.36e3T^{2} \)
41 \( 1 + (-2.66 + 4.62i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-33.9 + 19.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-9.64 + 5.56i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 35.8T + 2.80e3T^{2} \)
59 \( 1 + (20.8 + 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-37.9 - 65.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-31.8 - 18.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 87.8iT - 5.04e3T^{2} \)
73 \( 1 + 60.0T + 5.32e3T^{2} \)
79 \( 1 + (-32.1 + 18.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (66.0 - 38.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 27.5T + 7.92e3T^{2} \)
97 \( 1 + (-13.0 - 22.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−8.894219752700778713346136675902, −8.522242355135316319660181060951, −7.34527666814831558103605051899, −6.48513519176232101102795579598, −5.63639704605707142920543653046, −4.97146765202814535568574754726, −3.97478062555561719574574937932, −2.97237958466407108363120293881, −1.50354000524587270392901385842, −0.810394825176548256394342789308, 1.29183292968759141083709220295, 2.26494748873455647026973500464, 3.36726464977220866579248034517, 4.18026814969849220280965462665, 5.36625041275737731236570991188, 6.27717482506416998539497986661, 6.77834131299732864512695902780, 7.59655636455603902808335251378, 8.546090911132478190601816818300, 9.525669820935887900809986206864

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