Properties

Label 2-12e3-24.11-c3-0-6
Degree $2$
Conductor $1728$
Sign $-0.707 - 0.707i$
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  + 9.35·5-s − 21.2i·7-s + 50.9i·11-s − 33.2i·13-s + 120. i·17-s − 82.1·19-s + 95.3·23-s − 37.4·25-s + 83.1·29-s − 36.4i·31-s − 198. i·35-s − 201. i·37-s + 291. i·41-s − 457.·43-s − 628.·47-s + ⋯
L(s)  = 1  + 0.837·5-s − 1.14i·7-s + 1.39i·11-s − 0.709i·13-s + 1.72i·17-s − 0.991·19-s + 0.864·23-s − 0.299·25-s + 0.532·29-s − 0.210i·31-s − 0.958i·35-s − 0.896i·37-s + 1.11i·41-s − 1.62·43-s − 1.95·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8368785339\)
\(L(\frac12)\) \(\approx\) \(0.8368785339\)
\(L(\frac{5}{2})\) 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 \( 1 - 9.35T + 125T^{2} \)
7 \( 1 + 21.2iT - 343T^{2} \)
11 \( 1 - 50.9iT - 1.33e3T^{2} \)
13 \( 1 + 33.2iT - 2.19e3T^{2} \)
17 \( 1 - 120. iT - 4.91e3T^{2} \)
19 \( 1 + 82.1T + 6.85e3T^{2} \)
23 \( 1 - 95.3T + 1.21e4T^{2} \)
29 \( 1 - 83.1T + 2.43e4T^{2} \)
31 \( 1 + 36.4iT - 2.97e4T^{2} \)
37 \( 1 + 201. iT - 5.06e4T^{2} \)
41 \( 1 - 291. iT - 6.89e4T^{2} \)
43 \( 1 + 457.T + 7.95e4T^{2} \)
47 \( 1 + 628.T + 1.03e5T^{2} \)
53 \( 1 + 659.T + 1.48e5T^{2} \)
59 \( 1 + 44.8iT - 2.05e5T^{2} \)
61 \( 1 - 149. iT - 2.26e5T^{2} \)
67 \( 1 - 134.T + 3.00e5T^{2} \)
71 \( 1 + 291.T + 3.57e5T^{2} \)
73 \( 1 - 888.T + 3.89e5T^{2} \)
79 \( 1 + 714. iT - 4.93e5T^{2} \)
83 \( 1 - 827. iT - 5.71e5T^{2} \)
89 \( 1 + 19.6iT - 7.04e5T^{2} \)
97 \( 1 + 805.T + 9.12e5T^{2} \)
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

−9.480421626408609912768059672727, −8.330301521445360024960843866012, −7.73681884756762510941559354667, −6.71106458459353701902541232940, −6.25523155944935097426677061671, −5.07605426115823634650455908569, −4.34835282577419325527387606131, −3.41121994027112339912629064660, −2.07521186517975615602216602730, −1.34490550852125030756225633462, 0.16277547213795581016096174112, 1.58854002979042939330576217709, 2.58206503618285120451842799106, 3.30927200281234981102744210465, 4.81467328826673257685767153455, 5.37734770987870306124416533159, 6.30554622499860108585688270487, 6.76362938029008577356968178502, 8.126229176243342587979264722711, 8.768730263589721046884388713002

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