Properties

Label 2-12e3-12.11-c3-0-42
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $101.955$
Root an. cond. $10.0972$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5.83i·5-s + 8.83i·7-s − 23.6·11-s − 54.6·13-s − 117. i·17-s + 109. i·19-s − 33.5·23-s + 90.9·25-s − 40.0i·29-s − 292. i·31-s − 51.5·35-s − 283.·37-s + 367. i·41-s − 323. i·43-s + 66.2·47-s + ⋯
L(s)  = 1  + 0.522i·5-s + 0.476i·7-s − 0.647·11-s − 1.16·13-s − 1.67i·17-s + 1.32i·19-s − 0.304·23-s + 0.727·25-s − 0.256i·29-s − 1.69i·31-s − 0.249·35-s − 1.25·37-s + 1.39i·41-s − 1.14i·43-s + 0.205·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.517245914\)
\(L(\frac12)\) \(\approx\) \(1.517245914\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.83iT - 125T^{2} \)
7 \( 1 - 8.83iT - 343T^{2} \)
11 \( 1 + 23.6T + 1.33e3T^{2} \)
13 \( 1 + 54.6T + 2.19e3T^{2} \)
17 \( 1 + 117. iT - 4.91e3T^{2} \)
19 \( 1 - 109. iT - 6.85e3T^{2} \)
23 \( 1 + 33.5T + 1.21e4T^{2} \)
29 \( 1 + 40.0iT - 2.43e4T^{2} \)
31 \( 1 + 292. iT - 2.97e4T^{2} \)
37 \( 1 + 283.T + 5.06e4T^{2} \)
41 \( 1 - 367. iT - 6.89e4T^{2} \)
43 \( 1 + 323. iT - 7.95e4T^{2} \)
47 \( 1 - 66.2T + 1.03e5T^{2} \)
53 \( 1 + 158. iT - 1.48e5T^{2} \)
59 \( 1 - 848.T + 2.05e5T^{2} \)
61 \( 1 - 348.T + 2.26e5T^{2} \)
67 \( 1 - 194. iT - 3.00e5T^{2} \)
71 \( 1 - 939.T + 3.57e5T^{2} \)
73 \( 1 + 473.T + 3.89e5T^{2} \)
79 \( 1 - 273. iT - 4.93e5T^{2} \)
83 \( 1 + 338.T + 5.71e5T^{2} \)
89 \( 1 + 739. iT - 7.04e5T^{2} \)
97 \( 1 + 448.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.012985270046454003926910183491, −8.047721780210572364391018884307, −7.39006824756841688084144413857, −6.67991114119424003298592751431, −5.57009103870119087107549345878, −5.04785763783826690251876873694, −3.88116467000379093505281879990, −2.74918609990779434302988359839, −2.19467459758766865814649773854, −0.50516639902771721452694085964, 0.63539850038735781088529678807, 1.85323864361747824132731149838, 2.93173053243105515230139541697, 4.03878303303775294221280147378, 4.91737395594279879403977319724, 5.50625536914516075489903625973, 6.78991139528305802138125133276, 7.24751111856124415694003703535, 8.350989785938276676329298377451, 8.774764993507113655847494299656

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