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Properties

Label 2-432-9.7-c5-0-4
Degree $2$
Conductor $432$
Sign $-0.403 - 0.914i$
Analytic cond. $69.2858$
Root an. cond. $8.32380$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.1 − 22.7i)5-s + (31.6 − 54.7i)7-s + (−49.1 + 85.0i)11-s + (369. + 639. i)13-s − 250.·17-s − 1.10e3·19-s + (−2.20e3 − 3.81e3i)23-s + (1.21e3 − 2.10e3i)25-s + (−3.94e3 + 6.82e3i)29-s + (−2.30e3 − 3.99e3i)31-s − 1.66e3·35-s + 1.18e4·37-s + (5.04e3 + 8.73e3i)41-s + (3.51e3 − 6.09e3i)43-s + (−7.45e3 + 1.29e4i)47-s + ⋯
L(s)  = 1  + (−0.235 − 0.407i)5-s + (0.243 − 0.422i)7-s + (−0.122 + 0.211i)11-s + (0.605 + 1.04i)13-s − 0.209·17-s − 0.700·19-s + (−0.869 − 1.50i)23-s + (0.389 − 0.674i)25-s + (−0.870 + 1.50i)29-s + (−0.430 − 0.746i)31-s − 0.229·35-s + 1.42·37-s + (0.468 + 0.811i)41-s + (0.290 − 0.502i)43-s + (−0.492 + 0.853i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.403 - 0.914i$
Analytic conductor: \(69.2858\)
Root analytic conductor: \(8.32380\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :5/2),\ -0.403 - 0.914i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7884873313\)
\(L(\frac12)\) \(\approx\) \(0.7884873313\)
\(L(\frac{7}{2})\) 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 \( 1 + (13.1 + 22.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-31.6 + 54.7i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (49.1 - 85.0i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-369. - 639. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 250.T + 1.41e6T^{2} \)
19 \( 1 + 1.10e3T + 2.47e6T^{2} \)
23 \( 1 + (2.20e3 + 3.81e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (3.94e3 - 6.82e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (2.30e3 + 3.99e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.18e4T + 6.93e7T^{2} \)
41 \( 1 + (-5.04e3 - 8.73e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-3.51e3 + 6.09e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (7.45e3 - 1.29e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 2.24e4T + 4.18e8T^{2} \)
59 \( 1 + (5.40e3 + 9.36e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-594. + 1.02e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.95e4 - 5.12e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.43e4T + 1.80e9T^{2} \)
73 \( 1 + 5.30e4T + 2.07e9T^{2} \)
79 \( 1 + (1.86e4 - 3.23e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (6.04e4 - 1.04e5i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 9.78e4T + 5.58e9T^{2} \)
97 \( 1 + (5.33e4 - 9.24e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−10.79965827885709634320517594335, −9.671067081725979608414955247793, −8.752404584140118544688420960586, −8.001123648548505022089542715902, −6.88819316456769898628695675995, −6.02151273484462028028943899630, −4.57470000989531558485001164869, −4.06247630138831739543069487528, −2.41836938930390435857416719089, −1.17791919478887944220701411809, 0.19624659922341981276877376179, 1.75437259233197916932336706082, 3.04047806284770002283160689355, 4.03519757497200360260219858891, 5.44571862718463039984669991757, 6.13511731479541726414307176494, 7.45332878163377716602602169731, 8.129603581209081253292150334927, 9.133182301865990205900536943445, 10.12495492148298754540427517446

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