↑

Properties

Label 2-432-48.35-c1-0-25
Degree $2$
Conductor $432$
Sign $-0.844 + 0.536i$
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  + (−1.39 − 0.215i)2-s + (1.90 + 0.602i)4-s + (−1.33 − 1.33i)5-s + 0.400·7-s + (−2.53 − 1.25i)8-s + (1.58 + 2.15i)10-s + (0.0888 − 0.0888i)11-s + (−3.59 − 3.59i)13-s + (−0.559 − 0.0863i)14-s + (3.27 + 2.29i)16-s + 0.898i·17-s + (−3.16 + 3.16i)19-s + (−1.74 − 3.35i)20-s + (−0.143 + 0.105i)22-s − 7.09i·23-s + ⋯
L(s)  = 1  + (−0.988 − 0.152i)2-s + (0.953 + 0.301i)4-s + (−0.598 − 0.598i)5-s + 0.151·7-s + (−0.896 − 0.443i)8-s + (0.500 + 0.682i)10-s + (0.0267 − 0.0267i)11-s + (−0.998 − 0.998i)13-s + (−0.149 − 0.0230i)14-s + (0.818 + 0.574i)16-s + 0.217i·17-s + (−0.726 + 0.726i)19-s + (−0.390 − 0.750i)20-s + (−0.0305 + 0.0223i)22-s − 1.47i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.844 + 0.536i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.844 + 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109194 - 0.375475i\)
\(L(\frac12)\) \(\approx\) \(0.109194 - 0.375475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.215i)T \)
3 \( 1 \)
good5 \( 1 + (1.33 + 1.33i)T + 5iT^{2} \)
7 \( 1 - 0.400T + 7T^{2} \)
11 \( 1 + (-0.0888 + 0.0888i)T - 11iT^{2} \)
13 \( 1 + (3.59 + 3.59i)T + 13iT^{2} \)
17 \( 1 - 0.898iT - 17T^{2} \)
19 \( 1 + (3.16 - 3.16i)T - 19iT^{2} \)
23 \( 1 + 7.09iT - 23T^{2} \)
29 \( 1 + (5.95 - 5.95i)T - 29iT^{2} \)
31 \( 1 + 5.25iT - 31T^{2} \)
37 \( 1 + (-0.934 + 0.934i)T - 37iT^{2} \)
41 \( 1 + 5.47T + 41T^{2} \)
43 \( 1 + (6.81 + 6.81i)T + 43iT^{2} \)
47 \( 1 - 6.60T + 47T^{2} \)
53 \( 1 + (3.33 + 3.33i)T + 53iT^{2} \)
59 \( 1 + (-4.12 + 4.12i)T - 59iT^{2} \)
61 \( 1 + (7.11 + 7.11i)T + 61iT^{2} \)
67 \( 1 + (1.42 - 1.42i)T - 67iT^{2} \)
71 \( 1 - 0.567iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + 4.45iT - 79T^{2} \)
83 \( 1 + (-6.47 - 6.47i)T + 83iT^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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

−10.59109856919228117031313133931, −9.973404887469430427796620221081, −8.799509127517975334113695601242, −8.181032938127842271443534956784, −7.41448659374533267313853644996, −6.26044736263309349616583117276, −4.96366878212763825153928294388, −3.59600563154725244117396130460, −2.13443205926727692077712825914, −0.32048298625936228101615285211, 1.93264268301034025366463397586, 3.28922720887834364318852660087, 4.81969502252162894059048335248, 6.22909385851024466648131759613, 7.21417351337311328459675483013, 7.67255773014425667733349220877, 8.923008429903882397361117412100, 9.604930786274546590410604105345, 10.57401302080727703446737826962, 11.56718889399940091840866323580

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