Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $0.615 + 0.788i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.75 − 3.59i)2-s + (−9.84 − 12.6i)4-s − 46.6·5-s + 60.5i·7-s + (−62.6 + 13.2i)8-s + (−81.9 + 167. i)10-s − 73.6i·11-s − 31.1·13-s + (217. + 106. i)14-s + (−62.2 + 248. i)16-s − 53.8·17-s − 54.9i·19-s + (459. + 589. i)20-s + (−264. − 129. i)22-s − 281. i·23-s + ⋯
L(s)  = 1  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s − 1.86·5-s + 1.23i·7-s + (−0.978 + 0.206i)8-s + (−0.819 + 1.67i)10-s − 0.608i·11-s − 0.184·13-s + (1.11 + 0.542i)14-s + (−0.243 + 0.969i)16-s − 0.186·17-s − 0.152i·19-s + (1.14 + 1.47i)20-s + (−0.546 − 0.266i)22-s − 0.532i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $0.615 + 0.788i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ 0.615 + 0.788i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.126431091\)
\(L(\frac12)\)  \(\approx\)  \(1.126431091\)
\(L(3)\)   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 \( 1 + (-1.75 + 3.59i)T \)
3 \( 1 \)
good5 \( 1 + 46.6T + 625T^{2} \)
7 \( 1 - 60.5iT - 2.40e3T^{2} \)
11 \( 1 + 73.6iT - 1.46e4T^{2} \)
13 \( 1 + 31.1T + 2.85e4T^{2} \)
17 \( 1 + 53.8T + 8.35e4T^{2} \)
19 \( 1 + 54.9iT - 1.30e5T^{2} \)
23 \( 1 + 281. iT - 2.79e5T^{2} \)
29 \( 1 - 447.T + 7.07e5T^{2} \)
31 \( 1 - 277. iT - 9.23e5T^{2} \)
37 \( 1 - 1.01e3T + 1.87e6T^{2} \)
41 \( 1 + 1.89e3T + 2.82e6T^{2} \)
43 \( 1 + 769. iT - 3.41e6T^{2} \)
47 \( 1 + 2.74e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.64e3T + 7.89e6T^{2} \)
59 \( 1 + 303. iT - 1.21e7T^{2} \)
61 \( 1 - 956.T + 1.38e7T^{2} \)
67 \( 1 - 6.94e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.97e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.33e3T + 2.83e7T^{2} \)
79 \( 1 - 3.80e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.15e3iT - 4.74e7T^{2} \)
89 \( 1 - 7.13e3T + 6.27e7T^{2} \)
97 \( 1 + 1.96e3T + 8.85e7T^{2} \)
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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

−11.23149864927176014372359575170, −10.17916646898046059992652217321, −8.699454317533385190616126826317, −8.470884208430225847706835539115, −6.95110960180032164274104629210, −5.58512105729782787488778557321, −4.53464706301455528539952837134, −3.52313159457352292575470562517, −2.53383998771635565436146942111, −0.62829987318333312050513412956, 0.55698409499864168454470588774, 3.30455489427882110363233870011, 4.16789990100388753395711852496, 4.80022160512165283639684841009, 6.53203361270630694213757490349, 7.49573995117084742515630276489, 7.74305079079993066044654296061, 8.882199430164409728311590570471, 10.21892415070504738419356955246, 11.35802702778083431189953356343

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