↑

Properties

Label 2-432-16.13-c1-0-24
Degree $2$
Conductor $432$
Sign $0.955 - 0.295i$
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  + (1.15 + 0.812i)2-s + (0.678 + 1.88i)4-s + (2.39 − 2.39i)5-s − 2.42i·7-s + (−0.744 + 2.72i)8-s + (4.71 − 0.824i)10-s + (0.803 − 0.803i)11-s + (0.643 + 0.643i)13-s + (1.97 − 2.81i)14-s + (−3.07 + 2.55i)16-s + 0.717·17-s + (−3.76 − 3.76i)19-s + (6.12 + 2.87i)20-s + (1.58 − 0.276i)22-s + 7.35i·23-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)2-s + (0.339 + 0.940i)4-s + (1.07 − 1.07i)5-s − 0.918i·7-s + (−0.263 + 0.964i)8-s + (1.49 − 0.260i)10-s + (0.242 − 0.242i)11-s + (0.178 + 0.178i)13-s + (0.527 − 0.751i)14-s + (−0.769 + 0.638i)16-s + 0.174·17-s + (−0.864 − 0.864i)19-s + (1.37 + 0.643i)20-s + (0.337 − 0.0589i)22-s + 1.53i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.955 - 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47681 + 0.373666i\)
\(L(\frac12)\) \(\approx\) \(2.47681 + 0.373666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.812i)T \)
3 \( 1 \)
good5 \( 1 + (-2.39 + 2.39i)T - 5iT^{2} \)
7 \( 1 + 2.42iT - 7T^{2} \)
11 \( 1 + (-0.803 + 0.803i)T - 11iT^{2} \)
13 \( 1 + (-0.643 - 0.643i)T + 13iT^{2} \)
17 \( 1 - 0.717T + 17T^{2} \)
19 \( 1 + (3.76 + 3.76i)T + 19iT^{2} \)
23 \( 1 - 7.35iT - 23T^{2} \)
29 \( 1 + (-6.75 - 6.75i)T + 29iT^{2} \)
31 \( 1 + 6.69T + 31T^{2} \)
37 \( 1 + (7.02 - 7.02i)T - 37iT^{2} \)
41 \( 1 - 3.53iT - 41T^{2} \)
43 \( 1 + (-1.31 + 1.31i)T - 43iT^{2} \)
47 \( 1 + 4.02T + 47T^{2} \)
53 \( 1 + (-6.85 + 6.85i)T - 53iT^{2} \)
59 \( 1 + (4.81 - 4.81i)T - 59iT^{2} \)
61 \( 1 + (8.63 + 8.63i)T + 61iT^{2} \)
67 \( 1 + (-11.0 - 11.0i)T + 67iT^{2} \)
71 \( 1 + 8.42iT - 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 + 5.60T + 79T^{2} \)
83 \( 1 + (6.14 + 6.14i)T + 83iT^{2} \)
89 \( 1 + 5.44iT - 89T^{2} \)
97 \( 1 + 7.77T + 97T^{2} \)
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

−11.34524313792574387348583744603, −10.32448179943470936765528550670, −9.157647611024567490789535111660, −8.494187891372157217529530399453, −7.25391982840542609371284411005, −6.39917963863221394999804347468, −5.34839521650712794764005317738, −4.62254658090857386694191929089, −3.39259302786560582406253421837, −1.61695912842298359842504649647, 2.01666059279609606494333400534, 2.73337162312407199989111889348, 4.10172047359070650609555041757, 5.53883714370320502022479765894, 6.13965949151849457303371582856, 6.94366493548042838211061713181, 8.592265394843678160380406440533, 9.662895245960998998528884365951, 10.41022313758552113711169879155, 10.98514626163158224124306621853

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