↑

Properties

Label 2-432-27.22-c1-0-9
Degree $2$
Conductor $432$
Sign $0.734 - 0.678i$
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  + (1.54 + 0.782i)3-s + (2.29 − 0.836i)5-s + (0.775 + 4.39i)7-s + (1.77 + 2.41i)9-s + (−2.73 − 0.996i)11-s + (−2.01 − 1.69i)13-s + (4.20 + 0.505i)15-s + (−1.67 + 2.89i)17-s + (−1.02 − 1.77i)19-s + (−2.24 + 7.40i)21-s + (1.60 − 9.11i)23-s + (0.754 − 0.632i)25-s + (0.851 + 5.12i)27-s + (5.30 − 4.45i)29-s + (0.380 − 2.15i)31-s + ⋯
L(s)  = 1  + (0.892 + 0.451i)3-s + (1.02 − 0.374i)5-s + (0.293 + 1.66i)7-s + (0.591 + 0.806i)9-s + (−0.825 − 0.300i)11-s + (−0.559 − 0.469i)13-s + (1.08 + 0.130i)15-s + (−0.405 + 0.701i)17-s + (−0.234 − 0.406i)19-s + (−0.489 + 1.61i)21-s + (0.335 − 1.90i)23-s + (0.150 − 0.126i)25-s + (0.163 + 0.986i)27-s + (0.985 − 0.826i)29-s + (0.0683 − 0.387i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95375 + 0.764897i\)
\(L(\frac12)\) \(\approx\) \(1.95375 + 0.764897i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 - 0.782i)T \)
good5 \( 1 + (-2.29 + 0.836i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.775 - 4.39i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (2.73 + 0.996i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (2.01 + 1.69i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.02 + 1.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.60 + 9.11i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-5.30 + 4.45i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.380 + 2.15i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.13 - 2.63i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.42 + 1.61i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.03 - 5.89i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 1.97T + 53T^{2} \)
59 \( 1 + (-6.20 + 2.25i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.25 + 7.11i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.37 + 1.99i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-6.60 + 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.40 - 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.57 - 1.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.20 + 1.00i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (6.88 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.9 + 5.07i)T + (74.3 + 62.3i)T^{2} \)
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

−11.05513510135372979752438109777, −10.14558275735346681043713662602, −9.377719020757498225418443538576, −8.556274616030591327456990805321, −8.083118089510043943250594546091, −6.37254965857670498591022693867, −5.40447513841317744573545254997, −4.61845939782886963794948747004, −2.70456017742814604024780929516, −2.22398901487427558509126459106, 1.47152028118524127526515811537, 2.70065117957218662369690286180, 3.98954034476412813595337828587, 5.22631128103560143776106525252, 6.80208113478805974492020201486, 7.21842780023955193562559501154, 8.154574649855970056922481345913, 9.441189501532469119876874353859, 10.03116016094386396751442736575, 10.77947459162471084110606587867

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