Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $0.925 - 0.378i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.92 + 0.772i)2-s + (14.8 − 6.06i)4-s + 21.1·5-s − 44.6i·7-s + (−53.4 + 35.2i)8-s + (−83.0 + 16.3i)10-s + 67.7i·11-s − 29.1·13-s + (34.4 + 175. i)14-s + (182. − 179. i)16-s + 402.·17-s + 644. i·19-s + (313. − 128. i)20-s + (−52.3 − 265. i)22-s + 387. i·23-s + ⋯
L(s)  = 1  + (−0.981 + 0.193i)2-s + (0.925 − 0.378i)4-s + 0.846·5-s − 0.910i·7-s + (−0.834 + 0.550i)8-s + (−0.830 + 0.163i)10-s + 0.560i·11-s − 0.172·13-s + (0.175 + 0.893i)14-s + (0.712 − 0.701i)16-s + 1.39·17-s + 1.78i·19-s + (0.782 − 0.320i)20-s + (−0.108 − 0.549i)22-s + 0.732i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $0.925 - 0.378i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ 0.925 - 0.378i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.512908450\)
\(L(\frac12)\)  \(\approx\)  \(1.512908450\)
\(L(3)\)   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 \( 1 + (3.92 - 0.772i)T \)
3 \( 1 \)
good5 \( 1 - 21.1T + 625T^{2} \)
7 \( 1 + 44.6iT - 2.40e3T^{2} \)
11 \( 1 - 67.7iT - 1.46e4T^{2} \)
13 \( 1 + 29.1T + 2.85e4T^{2} \)
17 \( 1 - 402.T + 8.35e4T^{2} \)
19 \( 1 - 644. iT - 1.30e5T^{2} \)
23 \( 1 - 387. iT - 2.79e5T^{2} \)
29 \( 1 - 724.T + 7.07e5T^{2} \)
31 \( 1 + 1.25e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.40e3T + 1.87e6T^{2} \)
41 \( 1 - 1.54e3T + 2.82e6T^{2} \)
43 \( 1 + 1.87e3iT - 3.41e6T^{2} \)
47 \( 1 - 4.16e3iT - 4.87e6T^{2} \)
53 \( 1 - 906.T + 7.89e6T^{2} \)
59 \( 1 + 4.52e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.62e3T + 1.38e7T^{2} \)
67 \( 1 + 67.7iT - 2.01e7T^{2} \)
71 \( 1 - 1.31e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.47e3T + 2.83e7T^{2} \)
79 \( 1 - 4.36e3iT - 3.89e7T^{2} \)
83 \( 1 - 762. iT - 4.74e7T^{2} \)
89 \( 1 - 8.08e3T + 6.27e7T^{2} \)
97 \( 1 - 6.66e3T + 8.85e7T^{2} \)
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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

−10.59556888678352473913014408273, −9.974975178216399846525390489830, −9.479240465840368474299195116363, −8.011222490220829881337707446856, −7.46508095814868077258076686945, −6.26189012465455093144557120733, −5.42296848268964377790185229602, −3.67708362841177725382626774410, −2.05692598442237628055065984006, −0.997297513904881718713424443185, 0.793263006308184043272805299422, 2.21688548628047348791776390349, 3.13365787752053096416954896788, 5.19781437610346623828666686410, 6.15374455727541958530075572065, 7.13309978768917289618483569094, 8.432427372807083183868146940543, 9.017058209806212995455306431626, 9.929702470147813552391816700579, 10.69418071955392493769901458472

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