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Properties

Label 2-432-27.16-c1-0-12
Degree $2$
Conductor $432$
Sign $0.483 + 0.875i$
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  + (0.278 − 1.70i)3-s + (2.42 + 0.884i)5-s + (0.245 − 1.39i)7-s + (−2.84 − 0.950i)9-s + (1.99 − 0.725i)11-s + (1.59 − 1.33i)13-s + (2.18 − 3.90i)15-s + (1.93 + 3.35i)17-s + (−1.22 + 2.11i)19-s + (−2.31 − 0.807i)21-s + (−1.45 − 8.24i)23-s + (1.29 + 1.08i)25-s + (−2.41 + 4.59i)27-s + (−0.797 − 0.669i)29-s + (1.54 + 8.73i)31-s + ⋯
L(s)  = 1  + (0.160 − 0.987i)3-s + (1.08 + 0.395i)5-s + (0.0928 − 0.526i)7-s + (−0.948 − 0.316i)9-s + (0.600 − 0.218i)11-s + (0.441 − 0.370i)13-s + (0.564 − 1.00i)15-s + (0.469 + 0.813i)17-s + (−0.280 + 0.485i)19-s + (−0.505 − 0.176i)21-s + (−0.303 − 1.71i)23-s + (0.258 + 0.216i)25-s + (−0.465 + 0.885i)27-s + (−0.148 − 0.124i)29-s + (0.276 + 1.56i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50226 - 0.885848i\)
\(L(\frac12)\) \(\approx\) \(1.50226 - 0.885848i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.278 + 1.70i)T \)
good5 \( 1 + (-2.42 - 0.884i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.245 + 1.39i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.99 + 0.725i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.59 + 1.33i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.93 - 3.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.22 - 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.45 + 8.24i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.797 + 0.669i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.54 - 8.73i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (4.97 + 8.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.25 + 7.76i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (5.70 - 2.07i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.665 - 3.77i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 3.73T + 53T^{2} \)
59 \( 1 + (-5.04 - 1.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.85 - 10.5i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (11.3 - 9.50i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.68 + 4.64i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.25 - 9.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.10 - 5.12i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-9.32 - 7.82i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (7.20 - 12.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.6 + 3.87i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.76073578930226337168197600919, −10.35082360985421593174568312256, −9.020679329622305842097851604036, −8.280084332750466088010429469376, −7.15134330880968333451301279738, −6.30083654082567537478732308283, −5.67615449768378227754807101094, −3.89175850331913375476989309013, −2.51493361951274355199885090471, −1.29106943629347444353962486796, 1.87199613397715346973822172170, 3.28975003550784921746679532355, 4.60971965543247525516784141281, 5.50363230244232171263047826004, 6.30139799753018336148691567300, 7.82661921085094125618969551397, 9.036009503609863472299603781173, 9.426052626359518125067771199184, 10.11106996064494412093944265146, 11.38087491538268888653235027642

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