↑

Properties

Label 2-432-48.5-c0-0-1
Degree $2$
Conductor $432$
Sign $0.382 + 0.923i$
Analytic cond. $0.215596$
Root an. cond. $0.464323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)5-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·10-s + (−0.707 + 0.707i)11-s + (−1 + i)13-s + (0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (−0.707 − 0.707i)20-s + 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)5-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·10-s + (−0.707 + 0.707i)11-s + (−1 + i)13-s + (0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (−0.707 − 0.707i)20-s + 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(0.215596\)
Root analytic conductor: \(0.464323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216265391\)
\(L(\frac12)\) \(\approx\) \(1.216265391\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - T + T^{2} \)
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

−11.55293476886667045899638210271, −10.22037721964235543898145333497, −9.478054472216611856689181416419, −8.976497423871855767675419856033, −7.36313114954289662332727941628, −6.17306625928658912296584275359, −5.07640945539313864995736517812, −4.71388316553114415104045157819, −2.79685177606582793573701173399, −1.92571696198638370433387376998, 2.59097517581282865672808478937, 3.64572877319623334347323177999, 4.95121193538855816691733734501, 5.94129394461114701075032964697, 6.74305470865466984333616645250, 7.73768074330630743817850565657, 8.423625649063914643811153903773, 10.07811484228286977577933485556, 10.46200996594786302416744238994, 11.58706984603199617260118832604

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