Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-0.00268 - 0.999i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.82 − 2.83i)2-s + (−0.0429 − 15.9i)4-s + 11.7·5-s + 58.3i·7-s + (−45.4 − 45.0i)8-s + (33.2 − 33.3i)10-s + 100. i·11-s − 170.·13-s + (165. + 164. i)14-s + (−255. + 1.37i)16-s − 398.·17-s + 404. i·19-s + (−0.506 − 188. i)20-s + (284. + 283. i)22-s − 336. i·23-s + ⋯
L(s)  = 1  + (0.706 − 0.708i)2-s + (−0.00268 − 0.999i)4-s + 0.471·5-s + 1.19i·7-s + (−0.709 − 0.704i)8-s + (0.332 − 0.333i)10-s + 0.829i·11-s − 1.00·13-s + (0.843 + 0.841i)14-s + (−0.999 + 0.00536i)16-s − 1.37·17-s + 1.12i·19-s + (−0.00126 − 0.471i)20-s + (0.587 + 0.586i)22-s − 0.635i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00268 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00268 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-0.00268 - 0.999i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ -0.00268 - 0.999i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.123386471\)
\(L(\frac12)\)  \(\approx\)  \(1.123386471\)
\(L(3)\)   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 \( 1 + (-2.82 + 2.83i)T \)
3 \( 1 \)
good5 \( 1 - 11.7T + 625T^{2} \)
7 \( 1 - 58.3iT - 2.40e3T^{2} \)
11 \( 1 - 100. iT - 1.46e4T^{2} \)
13 \( 1 + 170.T + 2.85e4T^{2} \)
17 \( 1 + 398.T + 8.35e4T^{2} \)
19 \( 1 - 404. iT - 1.30e5T^{2} \)
23 \( 1 + 336. iT - 2.79e5T^{2} \)
29 \( 1 + 655.T + 7.07e5T^{2} \)
31 \( 1 - 635. iT - 9.23e5T^{2} \)
37 \( 1 + 1.59e3T + 1.87e6T^{2} \)
41 \( 1 - 2.46e3T + 2.82e6T^{2} \)
43 \( 1 + 2.23e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.90e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.29e3T + 7.89e6T^{2} \)
59 \( 1 - 1.15e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.92e3T + 1.38e7T^{2} \)
67 \( 1 + 3.56e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.63e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.49e3T + 2.83e7T^{2} \)
79 \( 1 + 3.22e3iT - 3.89e7T^{2} \)
83 \( 1 - 8.15e3iT - 4.74e7T^{2} \)
89 \( 1 - 910.T + 6.27e7T^{2} \)
97 \( 1 + 1.76e4T + 8.85e7T^{2} \)
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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

−11.38009328028221395176285534502, −10.30331032718226903214153882239, −9.550042031501181901025545348080, −8.729792532078679704330209902079, −7.12883132516000005030531034753, −5.99937112280177813143699566826, −5.19562889172416274031927248587, −4.11832125106079964664230729488, −2.50730432211703775738333266599, −1.90539997584294328743501911763, 0.23407322963278600661215194163, 2.35747476584839266914661880332, 3.75180197520940623210746638103, 4.71969096780956633421060834522, 5.80946303943961245852717957883, 6.89594480271707817216528277722, 7.52345769342103227870692116193, 8.734933790064241688512631779759, 9.703737676665155225710746938621, 10.96634905518558002187250010577

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