Properties

Label 2-12e3-9.2-c2-0-15
Degree $2$
Conductor $1728$
Sign $0.743 - 0.668i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−8.20 + 4.73i)5-s + (1.05 − 1.83i)7-s + (13.7 + 7.91i)11-s + (−4.70 − 8.14i)13-s − 11.6i·17-s − 12.9·19-s + (5.27 − 3.04i)23-s + (32.4 − 56.1i)25-s + (−24.7 − 14.2i)29-s + (8.75 + 15.1i)31-s + 20.0i·35-s + 15.6·37-s + (−14.8 + 8.54i)41-s + (21.7 − 37.6i)43-s + (−20.6 − 11.9i)47-s + ⋯
L(s)  = 1  + (−1.64 + 0.947i)5-s + (0.150 − 0.261i)7-s + (1.24 + 0.719i)11-s + (−0.361 − 0.626i)13-s − 0.682i·17-s − 0.682·19-s + (0.229 − 0.132i)23-s + (1.29 − 2.24i)25-s + (−0.854 − 0.493i)29-s + (0.282 + 0.489i)31-s + 0.572i·35-s + 0.422·37-s + (−0.361 + 0.208i)41-s + (0.505 − 0.874i)43-s + (−0.439 − 0.253i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.743 - 0.668i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.743 - 0.668i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.249473178\)
\(L(\frac12)\) \(\approx\) \(1.249473178\)
\(L(2)\) 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 \( 1 + (8.20 - 4.73i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-1.05 + 1.83i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-13.7 - 7.91i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4.70 + 8.14i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 11.6iT - 289T^{2} \)
19 \( 1 + 12.9T + 361T^{2} \)
23 \( 1 + (-5.27 + 3.04i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (24.7 + 14.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-8.75 - 15.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 15.6T + 1.36e3T^{2} \)
41 \( 1 + (14.8 - 8.54i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-21.7 + 37.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (20.6 + 11.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 14.1iT - 2.80e3T^{2} \)
59 \( 1 + (-38.5 + 22.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-1.86 + 3.22i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (21.0 + 36.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 120. iT - 5.04e3T^{2} \)
73 \( 1 - 5.48T + 5.32e3T^{2} \)
79 \( 1 + (-60.5 + 104. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-46.5 - 26.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 102. iT - 7.92e3T^{2} \)
97 \( 1 + (58.9 - 102. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−9.181736984122968586803077664803, −8.272608970980287606482364539814, −7.49998906423324265533665079640, −7.04736148595048774947893638186, −6.29270416942635528868504642066, −4.86571177282727794444385499716, −4.09899686989767632262998277053, −3.45045265044303906298965590567, −2.36624881950427896347338328540, −0.69834515623225336019608680188, 0.55290336676679586601001058861, 1.69540286260085326358642933717, 3.36117843499160567250211420311, 4.07881737448500816307567111665, 4.66434682912769943033815500486, 5.77894739633537050003130532587, 6.75280446418942999076328480002, 7.58464166919580164177615122467, 8.407879633405374805006484406773, 8.828174205004521295555184909061

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