↑

Properties

Label 2-12e3-144.59-c1-0-10
Degree $2$
Conductor $1728$
Sign $0.937 + 0.348i$
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  + (−0.178 − 0.664i)5-s + (−0.645 − 1.11i)7-s + (0.860 − 3.21i)11-s + (1.27 + 4.74i)13-s + 5.58i·17-s + (2.49 + 2.49i)19-s + (−2.36 − 1.36i)23-s + (3.92 − 2.26i)25-s + (0.792 − 2.95i)29-s + (5.28 + 3.04i)31-s + (−0.628 + 0.628i)35-s + (0.507 + 0.507i)37-s + (4.89 − 8.48i)41-s + (0.949 + 0.254i)43-s + (−6.13 − 10.6i)47-s + ⋯
L(s)  = 1  + (−0.0796 − 0.297i)5-s + (−0.244 − 0.422i)7-s + (0.259 − 0.968i)11-s + (0.352 + 1.31i)13-s + 1.35i·17-s + (0.572 + 0.572i)19-s + (−0.493 − 0.284i)23-s + (0.784 − 0.452i)25-s + (0.147 − 0.549i)29-s + (0.948 + 0.547i)31-s + (−0.106 + 0.106i)35-s + (0.0834 + 0.0834i)37-s + (0.765 − 1.32i)41-s + (0.144 + 0.0388i)43-s + (−0.895 − 1.55i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.937 + 0.348i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.705450534\)
\(L(\frac12)\) \(\approx\) \(1.705450534\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.178 + 0.664i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.645 + 1.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.860 + 3.21i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.27 - 4.74i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.58iT - 17T^{2} \)
19 \( 1 + (-2.49 - 2.49i)T + 19iT^{2} \)
23 \( 1 + (2.36 + 1.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.792 + 2.95i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-5.28 - 3.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.507 - 0.507i)T + 37iT^{2} \)
41 \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.949 - 0.254i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.13 + 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.601 - 0.601i)T - 53iT^{2} \)
59 \( 1 + (-4.77 + 1.28i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.8 - 2.90i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.110 - 0.0295i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.0447iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 + (-2.50 + 1.44i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.79 - 1.01i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-4.41 - 7.63i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.095810927910532935690953621938, −8.558432046654704412173737478521, −7.84393930396344867264056747138, −6.64942976364041252072373162907, −6.26627035359278547847194771056, −5.19505499172803168929344099300, −4.06065947885139910842472259896, −3.59260197765284572719769832233, −2.11169454977916017018025535164, −0.879117065913655694875192421330, 0.981024223967778207744905171525, 2.59376383147985472663395993630, 3.18284762073734775059092772208, 4.50206979621306520288807886332, 5.25904626456534743206067311899, 6.16769513574140269821476006598, 7.07007346626734448391472901611, 7.68421173142379392970195890930, 8.576789393047824050551595941414, 9.617589998234927044158949898660

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