Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} $
Sign $-0.948 + 0.316i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (55.1 + 95.6i)5-s + (50.8 − 88.1i)7-s + (75.1 − 130. i)11-s + (−317. − 550. i)13-s − 1.49e3·17-s − 1.43e3·19-s + (632. + 1.09e3i)23-s + (−4.53e3 + 7.84e3i)25-s + (−1.38e3 + 2.40e3i)29-s + (−3.48e3 − 6.03e3i)31-s + 1.12e4·35-s − 7.95e3·37-s + (−1.01e3 − 1.75e3i)41-s + (−6.26e3 + 1.08e4i)43-s + (3.24e3 − 5.61e3i)47-s + ⋯
L(s)  = 1  + (0.987 + 1.71i)5-s + (0.392 − 0.679i)7-s + (0.187 − 0.324i)11-s + (−0.521 − 0.903i)13-s − 1.25·17-s − 0.913·19-s + (0.249 + 0.431i)23-s + (−1.45 + 2.51i)25-s + (−0.306 + 0.531i)29-s + (−0.651 − 1.12i)31-s + 1.54·35-s − 0.954·37-s + (−0.0941 − 0.163i)41-s + (−0.516 + 0.894i)43-s + (0.214 − 0.370i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(432\)    =    \(2^{4} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.948 + 0.316i$
motivic weight  =  \(5\)
character  :  $\chi_{432} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 432,\ (\ :5/2),\ -0.948 + 0.316i)\)
\(L(3)\)  \(\approx\)  \(0.2816215007\)
\(L(\frac12)\)  \(\approx\)  \(0.2816215007\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-55.1 - 95.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-50.8 + 88.1i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-75.1 + 130. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (317. + 550. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.49e3T + 1.41e6T^{2} \)
19 \( 1 + 1.43e3T + 2.47e6T^{2} \)
23 \( 1 + (-632. - 1.09e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (1.38e3 - 2.40e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (3.48e3 + 6.03e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 7.95e3T + 6.93e7T^{2} \)
41 \( 1 + (1.01e3 + 1.75e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (6.26e3 - 1.08e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-3.24e3 + 5.61e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 9.82e3T + 4.18e8T^{2} \)
59 \( 1 + (2.35e4 + 4.07e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (4.16e3 - 7.22e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (3.63e3 + 6.28e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.58e3T + 1.80e9T^{2} \)
73 \( 1 - 5.80e4T + 2.07e9T^{2} \)
79 \( 1 + (3.18e4 - 5.52e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (4.14e4 - 7.17e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 3.86e3T + 5.58e9T^{2} \)
97 \( 1 + (3.46e4 - 5.99e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.94619472931988835575682262087, −10.11339410843896177412520110383, −9.253249655710360521734652729355, −7.902509196677383073545012767021, −6.99874249417149482696555960697, −6.34342539425703771729102194754, −5.28242236134276871645201027877, −3.82344726461034096339177613939, −2.74345498204683886897343746043, −1.77994726598425945062977259205, 0.05811273853293791908467591464, 1.63693066974209239542010839714, 2.19459029730057400675633366080, 4.35424094865553648356168209034, 4.91929077420871614336416771683, 5.88903002298725022172154674131, 6.91804271908197418349868258618, 8.497608875966789536667818597782, 8.872405270691687340257374414980, 9.561394010104843589694989329886

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