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  =  \((2,\ 324,\ (\ :2),\ 0.257 - 0.966i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.422489598\)
\(L(\frac12)\)  \(\approx\)  \(1.422489598\)
\(L(3)\)   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

−11.24590607850341999803311888942, −10.01077849513144014495628193287, −9.415520311317294072300784959950, −8.654760438544188999878122705095, −7.71457696523909535944582461822, −6.05369049601887161243805783656, −5.40565595717448463688893726965, −3.61724304697875629046577894873, −2.38831809629079742696581256196, −1.55829963531075217696718965759, 0.54065556130884804109463871060, 1.57325921024348215770459084389, 3.69787817404026090552634055983, 4.97812762470302982513364607197, 6.09777120421454798311425150599, 7.00715083358769427029177415303, 7.75645647002958399122169061806, 8.908136541729928847714432822160, 9.898767388914692802865219928014, 10.48057536040105690463126305577

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