Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-0.257 + 0.966i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.43 + 3.17i)2-s + (−4.11 + 15.4i)4-s − 22.1·5-s + 95.5i·7-s + (−59.0 + 24.6i)8-s + (−54.0 − 70.2i)10-s − 21.8i·11-s + 126.·13-s + (−303. + 233. i)14-s + (−222. − 127. i)16-s − 283.·17-s + 323. i·19-s + (91.0 − 342. i)20-s + (69.3 − 53.3i)22-s − 229. i·23-s + ⋯
L(s)  = 1  + (0.609 + 0.792i)2-s + (−0.257 + 0.966i)4-s − 0.885·5-s + 1.95i·7-s + (−0.922 + 0.385i)8-s + (−0.540 − 0.702i)10-s − 0.180i·11-s + 0.746·13-s + (−1.54 + 1.18i)14-s + (−0.867 − 0.496i)16-s − 0.982·17-s + 0.896i·19-s + (0.227 − 0.856i)20-s + (0.143 − 0.110i)22-s − 0.433i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-0.257 + 0.966i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ -0.257 + 0.966i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.8535999729\)
\(L(\frac12)\)  \(\approx\)  \(0.8535999729\)
\(L(3)\)   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 \( 1 + (-2.43 - 3.17i)T \)
3 \( 1 \)
good5 \( 1 + 22.1T + 625T^{2} \)
7 \( 1 - 95.5iT - 2.40e3T^{2} \)
11 \( 1 + 21.8iT - 1.46e4T^{2} \)
13 \( 1 - 126.T + 2.85e4T^{2} \)
17 \( 1 + 283.T + 8.35e4T^{2} \)
19 \( 1 - 323. iT - 1.30e5T^{2} \)
23 \( 1 + 229. iT - 2.79e5T^{2} \)
29 \( 1 - 1.20e3T + 7.07e5T^{2} \)
31 \( 1 + 829. iT - 9.23e5T^{2} \)
37 \( 1 + 318.T + 1.87e6T^{2} \)
41 \( 1 - 328.T + 2.82e6T^{2} \)
43 \( 1 + 207. iT - 3.41e6T^{2} \)
47 \( 1 + 1.22e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.83e3T + 7.89e6T^{2} \)
59 \( 1 + 1.47e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.87e3T + 1.38e7T^{2} \)
67 \( 1 + 247. iT - 2.01e7T^{2} \)
71 \( 1 + 4.30e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.01e3T + 2.83e7T^{2} \)
79 \( 1 - 7.19e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.32e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.54e3T + 6.27e7T^{2} \)
97 \( 1 - 5.83e3T + 8.85e7T^{2} \)
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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

−11.94947785676057817656285440586, −11.09485899283874428394500294012, −9.364949721352874688302451210166, −8.462531818742587492227316353292, −8.048577898604589707695581319941, −6.54257423866667934816357578440, −5.86108474630017327097855693998, −4.75952965552699647632178589835, −3.59508750022029400396155459478, −2.39727460151719727915908501105, 0.21912276138223230490591733842, 1.29918758836951178928226739615, 3.16012556737665008125377375509, 4.12934482443382692509611963958, 4.70384486678127707249112584827, 6.45904756069060735232881119092, 7.26200510672943273786238779052, 8.462450430282631583743973080991, 9.700387663707605557917602233323, 10.75327706077385135598896794860

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