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Properties

Label 2-432-48.35-c1-0-0
Degree $2$
Conductor $432$
Sign $-0.768 + 0.639i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.379 + 1.36i)2-s + (−1.71 + 1.03i)4-s + (−0.463 − 0.463i)5-s − 1.85·7-s + (−2.05 − 1.93i)8-s + (0.455 − 0.808i)10-s + (−3.73 + 3.73i)11-s + (−4.31 − 4.31i)13-s + (−0.705 − 2.53i)14-s + (1.85 − 3.54i)16-s + 6.55i·17-s + (−1.31 + 1.31i)19-s + (1.27 + 0.314i)20-s + (−6.50 − 3.67i)22-s − 0.727i·23-s + ⋯
L(s)  = 1  + (0.268 + 0.963i)2-s + (−0.855 + 0.517i)4-s + (−0.207 − 0.207i)5-s − 0.702·7-s + (−0.727 − 0.685i)8-s + (0.144 − 0.255i)10-s + (−1.12 + 1.12i)11-s + (−1.19 − 1.19i)13-s + (−0.188 − 0.676i)14-s + (0.464 − 0.885i)16-s + 1.59i·17-s + (−0.301 + 0.301i)19-s + (0.284 + 0.0702i)20-s + (−1.38 − 0.782i)22-s − 0.151i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.768 + 0.639i$
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: \((2,\ 432,\ (\ :1/2),\ -0.768 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115973 - 0.320460i\)
\(L(\frac12)\) \(\approx\) \(0.115973 - 0.320460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.379 - 1.36i)T \)
3 \( 1 \)
good5 \( 1 + (0.463 + 0.463i)T + 5iT^{2} \)
7 \( 1 + 1.85T + 7T^{2} \)
11 \( 1 + (3.73 - 3.73i)T - 11iT^{2} \)
13 \( 1 + (4.31 + 4.31i)T + 13iT^{2} \)
17 \( 1 - 6.55iT - 17T^{2} \)
19 \( 1 + (1.31 - 1.31i)T - 19iT^{2} \)
23 \( 1 + 0.727iT - 23T^{2} \)
29 \( 1 + (0.896 - 0.896i)T - 29iT^{2} \)
31 \( 1 - 5.25iT - 31T^{2} \)
37 \( 1 + (-3.98 + 3.98i)T - 37iT^{2} \)
41 \( 1 + 5.10T + 41T^{2} \)
43 \( 1 + (-1.09 - 1.09i)T + 43iT^{2} \)
47 \( 1 - 6.81T + 47T^{2} \)
53 \( 1 + (-0.712 - 0.712i)T + 53iT^{2} \)
59 \( 1 + (1.86 - 1.86i)T - 59iT^{2} \)
61 \( 1 + (-5.98 - 5.98i)T + 61iT^{2} \)
67 \( 1 + (8.48 - 8.48i)T - 67iT^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + 3.72iT - 73T^{2} \)
79 \( 1 - 0.199iT - 79T^{2} \)
83 \( 1 + (9.97 + 9.97i)T + 83iT^{2} \)
89 \( 1 - 7.53T + 89T^{2} \)
97 \( 1 - 9.81T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.24873003469288149651097631457, −10.30959513435012602816276790217, −10.06494200448427409562173521434, −8.707665840355130777689830127445, −7.87218977344772406519027126409, −7.17039304731476962778267121477, −6.02399649912944432249520597229, −5.11475886086131125232242620316, −4.11092041693343040007146157135, −2.72279340246640393945039389823, 0.18550372604527609068423645433, 2.42933629692112069830752500912, 3.26526990704040837655309002621, 4.61487930846027460021060859441, 5.53647523015801931346225780225, 6.80936356011709924996118848066, 7.932042470406747442819508948285, 9.250840415097510455235448484188, 9.659511628283473636152979038705, 10.79419122804267790360986364020

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