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Properties

Label 2-432-144.5-c2-0-40
Degree $2$
Conductor $432$
Sign $0.257 + 0.966i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 − 0.256i)2-s + (3.86 − 1.01i)4-s + (−1.00 − 3.74i)5-s + (4.65 − 2.68i)7-s + (7.41 − 3.00i)8-s + (−2.94 − 7.16i)10-s + (−12.4 − 3.34i)11-s + (−3.07 − 11.4i)13-s + (8.54 − 6.52i)14-s + (13.9 − 7.86i)16-s + 11.8i·17-s + (20.1 − 20.1i)19-s + (−7.68 − 13.4i)20-s + (−25.6 − 3.43i)22-s + (−4.74 + 8.20i)23-s + ⋯
L(s)  = 1  + (0.991 − 0.128i)2-s + (0.967 − 0.254i)4-s + (−0.200 − 0.748i)5-s + (0.664 − 0.383i)7-s + (0.926 − 0.375i)8-s + (−0.294 − 0.716i)10-s + (−1.13 − 0.304i)11-s + (−0.236 − 0.883i)13-s + (0.610 − 0.465i)14-s + (0.870 − 0.491i)16-s + 0.699i·17-s + (1.05 − 1.05i)19-s + (−0.384 − 0.672i)20-s + (−1.16 − 0.156i)22-s + (−0.206 + 0.356i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.257 + 0.966i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ 0.257 + 0.966i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.51299 - 1.93195i\)
\(L(\frac12)\) \(\approx\) \(2.51299 - 1.93195i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 + 0.256i)T \)
3 \( 1 \)
good5 \( 1 + (1.00 + 3.74i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (-4.65 + 2.68i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (12.4 + 3.34i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (3.07 + 11.4i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 - 11.8iT - 289T^{2} \)
19 \( 1 + (-20.1 + 20.1i)T - 361iT^{2} \)
23 \( 1 + (4.74 - 8.20i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (5.39 - 20.1i)T + (-728. - 420.5i)T^{2} \)
31 \( 1 + (19.3 - 33.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-9.65 - 9.65i)T + 1.36e3iT^{2} \)
41 \( 1 + (-11.7 + 20.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-19.5 + 72.8i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-18.4 + 10.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-21.4 + 21.4i)T - 2.80e3iT^{2} \)
59 \( 1 + (-6.02 - 22.4i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-58.9 - 15.7i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-24.1 - 90.1i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 124.T + 5.04e3T^{2} \)
73 \( 1 - 86.4iT - 5.32e3T^{2} \)
79 \( 1 + (-46.4 - 80.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (20.2 - 75.5i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 164.T + 7.92e3T^{2} \)
97 \( 1 + (-15.8 - 27.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.81862210518410639969613333109, −10.28331406888996226820654428222, −8.782401147548222822069586182986, −7.81785819524327234238324519919, −7.06449712368437587268270673259, −5.39671006134165792046006908538, −5.17038724020755633577170662786, −3.88769252061281350149758496643, −2.65203162761843190516175592548, −1.02297600695192337244800918603, 2.05083454594789564378800818973, 3.05725269955936919625320255399, 4.38102043005417274251137004810, 5.30787006655587293809602420127, 6.28461700560765065697741834294, 7.50611541945864753957102374193, 7.84262380141931419478474645823, 9.465017945593861704123208306487, 10.52108538338850476756528526146, 11.38990270584555199122053416831

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