Properties

Label 2-12e3-4.3-c2-0-57
Degree $2$
Conductor $1728$
Sign $-1$
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  − 6.42·5-s − 13.8i·7-s − 13.0i·11-s − 7.16·13-s + 31.5·17-s − 16.4i·19-s − 16.9i·23-s + 16.2·25-s + 42.4·29-s − 29.6i·31-s + 88.6i·35-s − 39.3·37-s + 39.8·41-s − 16.3i·43-s − 57.8i·47-s + ⋯
L(s)  = 1  − 1.28·5-s − 1.97i·7-s − 1.18i·11-s − 0.551·13-s + 1.85·17-s − 0.867i·19-s − 0.738i·23-s + 0.649·25-s + 1.46·29-s − 0.957i·31-s + 2.53i·35-s − 1.06·37-s + 0.972·41-s − 0.379i·43-s − 1.23i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.133859906\)
\(L(\frac12)\) \(\approx\) \(1.133859906\)
\(L(2)\) 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 \( 1 + 6.42T + 25T^{2} \)
7 \( 1 + 13.8iT - 49T^{2} \)
11 \( 1 + 13.0iT - 121T^{2} \)
13 \( 1 + 7.16T + 169T^{2} \)
17 \( 1 - 31.5T + 289T^{2} \)
19 \( 1 + 16.4iT - 361T^{2} \)
23 \( 1 + 16.9iT - 529T^{2} \)
29 \( 1 - 42.4T + 841T^{2} \)
31 \( 1 + 29.6iT - 961T^{2} \)
37 \( 1 + 39.3T + 1.36e3T^{2} \)
41 \( 1 - 39.8T + 1.68e3T^{2} \)
43 \( 1 + 16.3iT - 1.84e3T^{2} \)
47 \( 1 + 57.8iT - 2.20e3T^{2} \)
53 \( 1 + 46.4T + 2.80e3T^{2} \)
59 \( 1 - 14.2iT - 3.48e3T^{2} \)
61 \( 1 - 63.7T + 3.72e3T^{2} \)
67 \( 1 - 32.5iT - 4.48e3T^{2} \)
71 \( 1 + 22.4iT - 5.04e3T^{2} \)
73 \( 1 - 24.9T + 5.32e3T^{2} \)
79 \( 1 - 61.9iT - 6.24e3T^{2} \)
83 \( 1 - 44.7iT - 6.88e3T^{2} \)
89 \( 1 + 1.95T + 7.92e3T^{2} \)
97 \( 1 + 44.5T + 9.40e3T^{2} \)
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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

−8.396616289833407213730855000415, −7.939367238433302250974553271882, −7.25185353118116026558533376511, −6.62476574326755147194043331256, −5.32961194293462331536804130855, −4.34819918599531561834357477066, −3.72904659937625251600358799773, −2.98429995071267744654173145806, −0.923979759052119839117876720913, −0.40164512519473449376161214319, 1.52592501858461000623439795519, 2.74195701318936667850682720914, 3.51879626674420204093894745036, 4.73139002349061859527212889794, 5.35700249757361263240589631898, 6.26925379998407604038525483325, 7.40217911629543310803399993743, 7.943128411736821598429057959524, 8.608941513033665279249545502581, 9.566861090195244830587280942842

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