Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} $
Sign $-0.634 + 0.773i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−40.7 − 70.5i)5-s + (−89.6 + 155. i)7-s + (250. − 433. i)11-s + (275. + 476. i)13-s − 753.·17-s + 2.57e3·19-s + (1.37e3 + 2.37e3i)23-s + (−1.75e3 + 3.03e3i)25-s + (1.95e3 − 3.38e3i)29-s + (−1.55e3 − 2.68e3i)31-s + 1.46e4·35-s − 9.56e3·37-s + (1.11e3 + 1.92e3i)41-s + (7.14e3 − 1.23e4i)43-s + (3.23e3 − 5.60e3i)47-s + ⋯
L(s)  = 1  + (−0.728 − 1.26i)5-s + (−0.691 + 1.19i)7-s + (0.623 − 1.08i)11-s + (0.451 + 0.782i)13-s − 0.632·17-s + 1.63·19-s + (0.541 + 0.937i)23-s + (−0.561 + 0.972i)25-s + (0.431 − 0.747i)29-s + (−0.290 − 0.502i)31-s + 2.01·35-s − 1.14·37-s + (0.103 + 0.179i)41-s + (0.589 − 1.02i)43-s + (0.213 − 0.370i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(432\)    =    \(2^{4} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.634 + 0.773i$
motivic weight  =  \(5\)
character  :  $\chi_{432} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 432,\ (\ :5/2),\ -0.634 + 0.773i)\)
\(L(3)\)  \(\approx\)  \(0.9680371539\)
\(L(\frac12)\)  \(\approx\)  \(0.9680371539\)
\(L(\frac{7}{2})\)   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 \( 1 + (40.7 + 70.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (89.6 - 155. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-250. + 433. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-275. - 476. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 753.T + 1.41e6T^{2} \)
19 \( 1 - 2.57e3T + 2.47e6T^{2} \)
23 \( 1 + (-1.37e3 - 2.37e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-1.95e3 + 3.38e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (1.55e3 + 2.68e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 9.56e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.11e3 - 1.92e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-7.14e3 + 1.23e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-3.23e3 + 5.60e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 1.36e4T + 4.18e8T^{2} \)
59 \( 1 + (2.85e3 + 4.94e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-5.89e3 + 1.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.77e3 + 3.06e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 5.84e4T + 1.80e9T^{2} \)
73 \( 1 + 6.01e4T + 2.07e9T^{2} \)
79 \( 1 + (2.78e4 - 4.81e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.99e4 - 3.46e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 1.03e5T + 5.58e9T^{2} \)
97 \( 1 + (-8.29e4 + 1.43e5i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−9.661204353062837347270748584524, −8.913640014923340041310800240436, −8.591207095793577639064840981197, −7.29407451257422284842692257936, −6.05101564813990480237331777002, −5.30553609484535573593273145457, −4.07889698178167329772061109305, −3.09489101514373179837836975586, −1.45129281058604943554729223218, −0.27417812191098239816586120222, 1.09960398847546995842225922166, 2.94030850685471269005598771093, 3.61569023414453961686202447946, 4.67795899003804690456239062525, 6.31214828069617177766046816166, 7.18563778945574007907756425007, 7.43158997134518250230237698172, 8.914718370973754505233889107392, 10.08299344001563301826432813499, 10.56398154123260264160313032160

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