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Properties

Label 2-432-108.83-c1-0-12
Degree $2$
Conductor $432$
Sign $0.0323 + 0.999i$
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  + (−0.249 + 1.71i)3-s + (−1.47 − 1.75i)5-s + (−0.590 − 1.62i)7-s + (−2.87 − 0.854i)9-s + (0.867 + 0.728i)11-s + (−1.22 − 6.94i)13-s + (3.38 − 2.09i)15-s + (−1.48 + 0.857i)17-s + (−3.94 − 2.27i)19-s + (2.92 − 0.608i)21-s + (6.51 + 2.37i)23-s + (−0.0475 + 0.269i)25-s + (2.18 − 4.71i)27-s + (−6.35 − 1.12i)29-s + (1.56 − 4.30i)31-s + ⋯
L(s)  = 1  + (−0.143 + 0.989i)3-s + (−0.660 − 0.786i)5-s + (−0.223 − 0.613i)7-s + (−0.958 − 0.284i)9-s + (0.261 + 0.219i)11-s + (−0.339 − 1.92i)13-s + (0.873 − 0.540i)15-s + (−0.360 + 0.207i)17-s + (−0.904 − 0.522i)19-s + (0.639 − 0.132i)21-s + (1.35 + 0.494i)23-s + (−0.00950 + 0.0538i)25-s + (0.419 − 0.907i)27-s + (−1.17 − 0.208i)29-s + (0.281 − 0.773i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0323 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0323 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0323 + 0.999i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.0323 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522666 - 0.506009i\)
\(L(\frac12)\) \(\approx\) \(0.522666 - 0.506009i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.249 - 1.71i)T \)
good5 \( 1 + (1.47 + 1.75i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.590 + 1.62i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.867 - 0.728i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.22 + 6.94i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.48 - 0.857i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.94 + 2.27i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.51 - 2.37i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (6.35 + 1.12i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.56 + 4.30i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.47 - 2.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.14 + 1.25i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.337 - 0.402i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (8.97 - 3.26i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 8.57iT - 53T^{2} \)
59 \( 1 + (11.0 - 9.25i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.269 - 0.0979i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (5.85 - 1.03i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.36 - 2.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.58 - 2.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.41 + 1.65i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.731 + 4.14i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (5.83 + 3.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.21 + 2.69i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.86694332064711209451683596062, −10.09667778808112050813393471863, −9.168396118022320800318120572002, −8.310835978359908701929164248087, −7.40270445967156399738271144988, −5.96998433240520484452466798887, −4.91126009504041927081351801061, −4.14623093717478039479388664994, −3.03203782895687957010529776923, −0.45992030437201958810298602596, 1.91969191828435945766313973309, 3.12309675785696281445747454952, 4.54223915001020200027989669081, 6.04707241716501969417092734430, 6.80803338695062439120111499575, 7.42041284811471256525028488627, 8.671005335266558037367438077761, 9.300722536901324554921208095146, 10.95474343271966120765652534317, 11.32806057225992626896815040609

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