Properties

Label 2-432-108.95-c1-0-10
Degree $2$
Conductor $432$
Sign $0.979 + 0.199i$
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  + (−1.22 + 1.22i)3-s + (1.15 − 1.37i)5-s + (−0.0374 + 0.102i)7-s + (−0.00480 − 2.99i)9-s + (1.39 − 1.16i)11-s + (0.628 − 3.56i)13-s + (0.273 + 3.09i)15-s + (3.51 + 2.03i)17-s + (−0.846 + 0.488i)19-s + (−0.0803 − 0.171i)21-s + (5.93 − 2.16i)23-s + (0.309 + 1.75i)25-s + (3.68 + 3.66i)27-s + (0.912 − 0.160i)29-s + (0.185 + 0.508i)31-s + ⋯
L(s)  = 1  + (−0.706 + 0.707i)3-s + (0.515 − 0.614i)5-s + (−0.0141 + 0.0389i)7-s + (−0.00160 − 0.999i)9-s + (0.419 − 0.352i)11-s + (0.174 − 0.988i)13-s + (0.0705 + 0.799i)15-s + (0.852 + 0.492i)17-s + (−0.194 + 0.112i)19-s + (−0.0175 − 0.0375i)21-s + (1.23 − 0.450i)23-s + (0.0618 + 0.350i)25-s + (0.708 + 0.705i)27-s + (0.169 − 0.0298i)29-s + (0.0332 + 0.0913i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.979 + 0.199i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23502 - 0.124674i\)
\(L(\frac12)\) \(\approx\) \(1.23502 - 0.124674i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.22i)T \)
good5 \( 1 + (-1.15 + 1.37i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.0374 - 0.102i)T + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.39 + 1.16i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.628 + 3.56i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.51 - 2.03i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.846 - 0.488i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.93 + 2.16i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.912 + 0.160i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.185 - 0.508i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-4.79 + 8.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0246 - 0.00434i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.93 + 8.25i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-7.54 - 2.74i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + (-1.98 - 1.66i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.80 + 2.11i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (13.5 + 2.39i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.650 + 1.12i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.34 + 5.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.71 - 1.53i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (3.02 + 17.1i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (4.27 - 2.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.6 - 8.91i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.86643352025455197608417078357, −10.40868133991711994995904777489, −9.296853802966634788386834465604, −8.728464255765027512640770044467, −7.37567000646452572115646186659, −5.99611760160292194769015732082, −5.52000305131785547783702614892, −4.39777979697522654823538568777, −3.18148748252161103523562040318, −1.04037712848531684433911299793, 1.42835552225765145950531735605, 2.81194922589134198919891223661, 4.51632024862067896016006353067, 5.63661687735718882591371264231, 6.66121567400877865938242490644, 7.10562391678807612545627211951, 8.344680740863200637012484200847, 9.562423743218919341673237224628, 10.33757895275652422707179707837, 11.39679080499808501329318742974

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