Properties

Label 2-432-108.83-c1-0-15
Degree $2$
Conductor $432$
Sign $-0.849 + 0.527i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
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  + (0.249 − 1.71i)3-s + (−1.47 − 1.75i)5-s + (0.590 + 1.62i)7-s + (−2.87 − 0.854i)9-s + (−0.867 − 0.728i)11-s + (−1.22 − 6.94i)13-s + (−3.38 + 2.09i)15-s + (−1.48 + 0.857i)17-s + (3.94 + 2.27i)19-s + (2.92 − 0.608i)21-s + (−6.51 − 2.37i)23-s + (−0.0475 + 0.269i)25-s + (−2.18 + 4.71i)27-s + (−6.35 − 1.12i)29-s + (−1.56 + 4.30i)31-s + ⋯
L(s)  = 1  + (0.143 − 0.989i)3-s + (−0.660 − 0.786i)5-s + (0.223 + 0.613i)7-s + (−0.958 − 0.284i)9-s + (−0.261 − 0.219i)11-s + (−0.339 − 1.92i)13-s + (−0.873 + 0.540i)15-s + (−0.360 + 0.207i)17-s + (0.904 + 0.522i)19-s + (0.639 − 0.132i)21-s + (−1.35 − 0.494i)23-s + (−0.00950 + 0.0538i)25-s + (−0.419 + 0.907i)27-s + (−1.17 − 0.208i)29-s + (−0.281 + 0.773i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.849 + 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260824 - 0.913957i\)
\(L(\frac12)\) \(\approx\) \(0.260824 - 0.913957i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.249 + 1.71i)T \)
good5 \( 1 + (1.47 + 1.75i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.590 - 1.62i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.867 + 0.728i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.22 + 6.94i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.48 - 0.857i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.94 - 2.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.51 + 2.37i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (6.35 + 1.12i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.56 - 4.30i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.47 - 2.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.14 + 1.25i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.337 + 0.402i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-8.97 + 3.26i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 8.57iT - 53T^{2} \)
59 \( 1 + (-11.0 + 9.25i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.269 - 0.0979i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-5.85 + 1.03i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (1.36 + 2.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.58 - 2.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.41 - 1.65i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.731 - 4.14i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (5.83 + 3.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.21 + 2.69i)T + (16.8 + 95.5i)T^{2} \)
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

−10.98362442503651308857106492358, −9.812395646452195012806136785030, −8.502029782517688287497650748150, −8.152024954105793966756159845358, −7.34419898502291063546949442908, −5.86456625247737370737543616128, −5.26314957371030454833940929279, −3.62779569119362482047988939359, −2.30617181039407182548157520253, −0.57887215987334659179598716001, 2.41592029664698926080371756660, 3.87321010254664929864932239092, 4.35964452013942080793406141090, 5.74422306379747635888204060208, 7.14600948156957838897011356054, 7.66294954810227016654344322383, 9.118208214648899951083261024632, 9.636665696990484654981868978957, 10.78116417873348651299609499866, 11.32306733463016500146424815314

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