Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $0.986 + 0.163i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.328 − 3.98i)2-s + (−15.7 − 2.61i)4-s − 5.66·5-s − 52.1i·7-s + (−15.6 + 62.0i)8-s + (−1.85 + 22.5i)10-s + 106. i·11-s − 122.·13-s + (−207. − 17.1i)14-s + (242. + 82.6i)16-s + 122.·17-s + 593. i·19-s + (89.3 + 14.8i)20-s + (425. + 35.0i)22-s − 546. i·23-s + ⋯
L(s)  = 1  + (0.0820 − 0.996i)2-s + (−0.986 − 0.163i)4-s − 0.226·5-s − 1.06i·7-s + (−0.244 + 0.969i)8-s + (−0.0185 + 0.225i)10-s + 0.881i·11-s − 0.722·13-s + (−1.06 − 0.0873i)14-s + (0.946 + 0.322i)16-s + 0.424·17-s + 1.64i·19-s + (0.223 + 0.0370i)20-s + (0.878 + 0.0723i)22-s − 1.03i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $0.986 + 0.163i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ 0.986 + 0.163i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.216625967\)
\(L(\frac12)\)  \(\approx\)  \(1.216625967\)
\(L(3)\)   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 \( 1 + (-0.328 + 3.98i)T \)
3 \( 1 \)
good5 \( 1 + 5.66T + 625T^{2} \)
7 \( 1 + 52.1iT - 2.40e3T^{2} \)
11 \( 1 - 106. iT - 1.46e4T^{2} \)
13 \( 1 + 122.T + 2.85e4T^{2} \)
17 \( 1 - 122.T + 8.35e4T^{2} \)
19 \( 1 - 593. iT - 1.30e5T^{2} \)
23 \( 1 + 546. iT - 2.79e5T^{2} \)
29 \( 1 + 735.T + 7.07e5T^{2} \)
31 \( 1 - 585. iT - 9.23e5T^{2} \)
37 \( 1 - 2.28e3T + 1.87e6T^{2} \)
41 \( 1 - 2.86e3T + 2.82e6T^{2} \)
43 \( 1 - 2.24e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.05e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.75e3T + 7.89e6T^{2} \)
59 \( 1 - 2.15e3iT - 1.21e7T^{2} \)
61 \( 1 + 66.3T + 1.38e7T^{2} \)
67 \( 1 - 4.10e3iT - 2.01e7T^{2} \)
71 \( 1 + 5.03e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.70e3T + 2.83e7T^{2} \)
79 \( 1 - 1.38e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.00e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.18e3T + 6.27e7T^{2} \)
97 \( 1 + 4.81e3T + 8.85e7T^{2} \)
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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

−10.86877341263773096506908193514, −10.09324664514895675798271073564, −9.526007236270379521308834800867, −8.079277245458877420066723085593, −7.37517352915212696385288518911, −5.82215760527688793048051846209, −4.47725343283274126446831039188, −3.82885337012658591817260011196, −2.34138684902191240460494630201, −1.00246280131063610279797108478, 0.44556546546620039477339252629, 2.65306890790590029806897270373, 4.03955558164876065744437145598, 5.35418175924442713428761314066, 5.93848291301766152401258815705, 7.23485547637445085067073116002, 8.019896194454010819744103667958, 9.099408236082279131903611741954, 9.578664926974631035262322781286, 11.15887575093082156720965824001

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