Properties

Label 2-432-9.5-c4-0-2
Degree $2$
Conductor $432$
Sign $-0.950 - 0.310i$
Analytic cond. $44.6558$
Root an. cond. $6.68250$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (34.8 + 20.1i)5-s + (7.38 + 12.7i)7-s + (−70.7 + 40.8i)11-s + (−139. + 240. i)13-s + 10.8i·17-s − 532.·19-s + (−702. − 405. i)23-s + (498. + 862. i)25-s + (257. − 148. i)29-s + (97.5 − 168. i)31-s + 594. i·35-s − 2.09e3·37-s + (1.35e3 + 784. i)41-s + (−46.0 − 79.8i)43-s + (1.84e3 − 1.06e3i)47-s + ⋯
L(s)  = 1  + (1.39 + 0.805i)5-s + (0.150 + 0.261i)7-s + (−0.584 + 0.337i)11-s + (−0.822 + 1.42i)13-s + 0.0376i·17-s − 1.47·19-s + (−1.32 − 0.766i)23-s + (0.797 + 1.38i)25-s + (0.306 − 0.176i)29-s + (0.101 − 0.175i)31-s + 0.485i·35-s − 1.53·37-s + (0.808 + 0.466i)41-s + (−0.0249 − 0.0431i)43-s + (0.837 − 0.483i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.950 - 0.310i$
Analytic conductor: \(44.6558\)
Root analytic conductor: \(6.68250\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :2),\ -0.950 - 0.310i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.200657845\)
\(L(\frac12)\) \(\approx\) \(1.200657845\)
\(L(3)\) 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 \( 1 + (-34.8 - 20.1i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (-7.38 - 12.7i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (70.7 - 40.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (139. - 240. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 10.8iT - 8.35e4T^{2} \)
19 \( 1 + 532.T + 1.30e5T^{2} \)
23 \( 1 + (702. + 405. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-257. + 148. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-97.5 + 168. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 2.09e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.35e3 - 784. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (46.0 + 79.8i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.84e3 + 1.06e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 2.57e3iT - 7.89e6T^{2} \)
59 \( 1 + (-1.34e3 - 778. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.68e3 + 4.65e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-457. + 792. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 8.21e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.43e3T + 2.83e7T^{2} \)
79 \( 1 + (-2.31e3 - 4.01e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (5.19e3 - 2.99e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 8.43e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.01e3 - 5.22e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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

−10.66067451434229590686043718809, −10.11622108364726614041647719063, −9.315959234798560175607559237924, −8.295374816725494726321042642925, −6.93792092282829413151426997764, −6.38102113930693295495921283644, −5.32430240265256889107840530437, −4.17297965080608579157095108246, −2.36050951510487719845651762177, −2.04886023275448708489715419670, 0.28476221209803961380903078143, 1.68388820194204323760240961115, 2.76995797454045081499300618463, 4.42944355473454224648502034832, 5.48811113292159639594576252583, 5.99953189485167259415725700880, 7.45664283670125854001925145447, 8.393046402196321994732290982218, 9.250846653055839566089467826581, 10.32063712247578763905978357369

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