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Properties

Label 2-432-144.11-c1-0-20
Degree $2$
Conductor $432$
Sign $-0.863 - 0.503i$
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  + (0.354 − 1.36i)2-s + (−1.74 − 0.970i)4-s + (−1.76 − 0.473i)5-s + (1.40 − 2.43i)7-s + (−1.94 + 2.04i)8-s + (−1.27 + 2.25i)10-s + (−5.79 + 1.55i)11-s + (−1.10 − 0.296i)13-s + (−2.83 − 2.78i)14-s + (2.11 + 3.39i)16-s − 0.699i·17-s + (−2.01 − 2.01i)19-s + (2.62 + 2.54i)20-s + (0.0708 + 8.48i)22-s + (4.91 − 2.83i)23-s + ⋯
L(s)  = 1  + (0.250 − 0.968i)2-s + (−0.874 − 0.485i)4-s + (−0.789 − 0.211i)5-s + (0.531 − 0.920i)7-s + (−0.689 + 0.724i)8-s + (−0.403 + 0.711i)10-s + (−1.74 + 0.468i)11-s + (−0.307 − 0.0823i)13-s + (−0.757 − 0.745i)14-s + (0.528 + 0.848i)16-s − 0.169i·17-s + (−0.462 − 0.462i)19-s + (0.587 + 0.568i)20-s + (0.0151 + 1.80i)22-s + (1.02 − 0.591i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.863 - 0.503i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.863 - 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155817 + 0.576490i\)
\(L(\frac12)\) \(\approx\) \(0.155817 + 0.576490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.354 + 1.36i)T \)
3 \( 1 \)
good5 \( 1 + (1.76 + 0.473i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.40 + 2.43i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.79 - 1.55i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.10 + 0.296i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 0.699iT - 17T^{2} \)
19 \( 1 + (2.01 + 2.01i)T + 19iT^{2} \)
23 \( 1 + (-4.91 + 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.41 - 1.45i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.15 - 2.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.48 - 3.48i)T + 37iT^{2} \)
41 \( 1 + (1.55 + 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.90 + 7.09i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.45 + 4.45i)T - 53iT^{2} \)
59 \( 1 + (-3.65 + 13.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.54 - 13.2i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.45 + 12.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.21iT - 71T^{2} \)
73 \( 1 + 3.75iT - 73T^{2} \)
79 \( 1 + (2.96 + 1.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.308 + 1.15i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 0.391T + 89T^{2} \)
97 \( 1 + (0.875 - 1.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.71882649191503414730434267423, −10.12453371082517365294081077158, −8.862336104333433738745364851651, −7.939287287809377736754019087084, −7.14333381147373405879748911318, −5.27882948874064792185777950739, −4.66268456599270794574094279809, −3.57863994353980330614838995370, −2.22685927465231035096073229439, −0.33712231872789589067796324323, 2.70305543734366114617048049190, 4.00448457083663075147992289117, 5.26167164991145272147769950321, 5.79024999244937036630492297212, 7.30203471181479479481054753616, 7.88291535741057307503736544135, 8.607404647895149192440837750196, 9.646660728055137346754580683606, 10.94400502368926315131441350831, 11.68515012405075484417878750849

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