Properties

Label 2-12e3-36.11-c1-0-18
Degree $2$
Conductor $1728$
Sign $-0.315 + 0.948i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.686 − 0.396i)5-s + (−2.35 − 1.35i)7-s + (1.71 − 2.96i)11-s + (1.68 + 2.92i)13-s − 2.52i·17-s − 2.20i·19-s + (−1.07 − 1.86i)23-s + (−2.18 + 3.78i)25-s + (−0.686 − 0.396i)29-s + (1.47 − 0.852i)31-s − 2.15·35-s − 4.74·37-s + (−0.127 + 0.0737i)41-s + (−6.01 − 3.47i)43-s + (5.77 − 10.0i)47-s + ⋯
L(s)  = 1  + (0.306 − 0.177i)5-s + (−0.888 − 0.513i)7-s + (0.516 − 0.894i)11-s + (0.467 + 0.809i)13-s − 0.612i·17-s − 0.506i·19-s + (−0.224 − 0.388i)23-s + (−0.437 + 0.757i)25-s + (−0.127 − 0.0735i)29-s + (0.265 − 0.153i)31-s − 0.363·35-s − 0.780·37-s + (−0.0199 + 0.0115i)41-s + (−0.917 − 0.529i)43-s + (0.842 − 1.45i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.315 + 0.948i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.226134060\)
\(L(\frac12)\) \(\approx\) \(1.226134060\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.686 + 0.396i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.35 + 1.35i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.71 + 2.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.68 - 2.92i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.52iT - 17T^{2} \)
19 \( 1 + 2.20iT - 19T^{2} \)
23 \( 1 + (1.07 + 1.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.686 + 0.396i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.47 + 0.852i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.74T + 37T^{2} \)
41 \( 1 + (0.127 - 0.0737i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.01 + 3.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.77 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 + (2.58 + 4.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.68 + 2.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.01 - 3.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.75T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (8.80 + 5.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.62 + 6.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.34iT - 89T^{2} \)
97 \( 1 + (-6.24 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.034251545939564099600833664446, −8.553144949602942013146520082346, −7.30331345034780569747561728956, −6.67406600658395408936650374411, −5.98189436820003186702581603015, −5.00238543802349077931194227339, −3.88074580891553822842346961778, −3.23905772667244709055868097554, −1.86230638476041415293179096631, −0.46203918409998538061410207575, 1.48589992828108610432835399879, 2.66786393125838756279043477595, 3.59059415980102795349685980664, 4.55410467511192062315887300457, 5.83093027994524958669537460320, 6.15171447003994979689144866285, 7.12452729400424943769746656440, 8.005354168489266556848143982087, 8.853366595478387953098471361294, 9.639126463963952580556531102686

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