Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-0.829 - 0.557i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.16 + 3.82i)2-s + (−13.2 − 8.92i)4-s − 39.0·5-s + 12.2i·7-s + (49.6 − 40.4i)8-s + (45.5 − 149. i)10-s − 111. i·11-s + 208.·13-s + (−46.6 − 14.2i)14-s + (96.6 + 237. i)16-s + 93.3·17-s + 26.8i·19-s + (518. + 348. i)20-s + (424. + 129. i)22-s − 874. i·23-s + ⋯
L(s)  = 1  + (−0.291 + 0.956i)2-s + (−0.829 − 0.557i)4-s − 1.56·5-s + 0.249i·7-s + (0.775 − 0.631i)8-s + (0.455 − 1.49i)10-s − 0.917i·11-s + 1.23·13-s + (−0.238 − 0.0726i)14-s + (0.377 + 0.925i)16-s + 0.323·17-s + 0.0744i·19-s + (1.29 + 0.871i)20-s + (0.877 + 0.267i)22-s − 1.65i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-0.829 - 0.557i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ -0.829 - 0.557i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.5950543510\)
\(L(\frac12)\)  \(\approx\)  \(0.5950543510\)
\(L(3)\)   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 \( 1 + (1.16 - 3.82i)T \)
3 \( 1 \)
good5 \( 1 + 39.0T + 625T^{2} \)
7 \( 1 - 12.2iT - 2.40e3T^{2} \)
11 \( 1 + 111. iT - 1.46e4T^{2} \)
13 \( 1 - 208.T + 2.85e4T^{2} \)
17 \( 1 - 93.3T + 8.35e4T^{2} \)
19 \( 1 - 26.8iT - 1.30e5T^{2} \)
23 \( 1 + 874. iT - 2.79e5T^{2} \)
29 \( 1 + 1.30e3T + 7.07e5T^{2} \)
31 \( 1 - 685. iT - 9.23e5T^{2} \)
37 \( 1 + 1.76e3T + 1.87e6T^{2} \)
41 \( 1 - 78.0T + 2.82e6T^{2} \)
43 \( 1 - 1.62e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.30e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.31e3T + 7.89e6T^{2} \)
59 \( 1 - 5.56e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.18e3T + 1.38e7T^{2} \)
67 \( 1 + 246. iT - 2.01e7T^{2} \)
71 \( 1 - 4.60e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.56e3T + 2.83e7T^{2} \)
79 \( 1 + 5.18e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.87e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.16e3T + 6.27e7T^{2} \)
97 \( 1 + 5.73e3T + 8.85e7T^{2} \)
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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

−11.18789394623724599437896253620, −10.54148547505496478908374109911, −8.965092590468941751034039677178, −8.443660588693679620645356308695, −7.67168642921440489683274086347, −6.62579015282706520480074541272, −5.61440430358903961577511935025, −4.30611477837558441478750673746, −3.42206849081103781550917391447, −0.923432797766425778454994357770, 0.27717247160335908242355890941, 1.70648808267101804980002262194, 3.55948704312845668881271806081, 3.91496662072985611375455085990, 5.26983275649394613466456284443, 7.18669956585527689689576772536, 7.82988148155087028023314516255, 8.768553426185071825496731658559, 9.755681680724063660787628415290, 10.86846724409766644689721759457

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