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Properties

Label 2-432-36.11-c3-0-12
Degree $2$
Conductor $432$
Sign $0.902 + 0.431i$
Analytic cond. $25.4888$
Root an. cond. $5.04864$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16.1 − 9.30i)5-s + (10.2 + 5.89i)7-s + (30.2 − 52.4i)11-s + (43.2 + 74.8i)13-s + 49.3i·17-s − 17.5i·19-s + (−50.2 − 86.9i)23-s + (110. − 191. i)25-s + (41.5 + 24.0i)29-s + (−227. + 131. i)31-s + 219.·35-s + 143.·37-s + (2.47 − 1.42i)41-s + (79.0 + 45.6i)43-s + (−65.9 + 114. i)47-s + ⋯
L(s)  = 1  + (1.44 − 0.832i)5-s + (0.551 + 0.318i)7-s + (0.830 − 1.43i)11-s + (0.922 + 1.59i)13-s + 0.704i·17-s − 0.211i·19-s + (−0.455 − 0.788i)23-s + (0.886 − 1.53i)25-s + (0.266 + 0.153i)29-s + (−1.31 + 0.759i)31-s + 1.06·35-s + 0.636·37-s + (0.00942 − 0.00544i)41-s + (0.280 + 0.161i)43-s + (−0.204 + 0.354i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.902 + 0.431i$
Analytic conductor: \(25.4888\)
Root analytic conductor: \(5.04864\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :3/2),\ 0.902 + 0.431i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.932093794\)
\(L(\frac12)\) \(\approx\) \(2.932093794\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + (-16.1 + 9.30i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-10.2 - 5.89i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-30.2 + 52.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-43.2 - 74.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 49.3iT - 4.91e3T^{2} \)
19 \( 1 + 17.5iT - 6.85e3T^{2} \)
23 \( 1 + (50.2 + 86.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-41.5 - 24.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (227. - 131. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 143.T + 5.06e4T^{2} \)
41 \( 1 + (-2.47 + 1.42i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-79.0 - 45.6i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (65.9 - 114. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 187. iT - 1.48e5T^{2} \)
59 \( 1 + (119. + 207. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-221. + 383. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-266. + 153. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 402.T + 3.57e5T^{2} \)
73 \( 1 - 857.T + 3.89e5T^{2} \)
79 \( 1 + (458. + 264. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-202. + 351. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 238. iT - 7.04e5T^{2} \)
97 \( 1 + (-72.5 + 125. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.78202984747419859592062632700, −9.491849562825137135887222493892, −8.876550312191567386768807916450, −8.369089106445945536652492080191, −6.49895980766201343699828381904, −6.04685101945413709279120109050, −4.98307314946517967571420436615, −3.79719819341019566857064892947, −2.01873148267139372002755195047, −1.18599876068755395405900509412, 1.34351407486945787427917158554, 2.41597675982815840986780657168, 3.78198038404013609405727579430, 5.22890966260958884525157248373, 6.04289803136616394382566621488, 7.03728458859358815597535768065, 7.895109195577828770092208591395, 9.307914607821197501758155851650, 9.909369103355174345993138144098, 10.66084587847460935218252324302

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