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  =  \((2,\ 324,\ (\ :2),\ 0.00268 + 0.999i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.2169614586\)
\(L(\frac12)\)  \(\approx\)  \(0.2169614586\)
\(L(3)\)   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

−10.43281869562090574807366284960, −9.733702800463484506851849378990, −8.625081393164248471067583004129, −8.059288104498650537725733033098, −7.02678625197484867477900969411, −5.76148869484217812864887347073, −5.20768935565961040876308716306, −3.37850322052158848166775486409, −1.78247774284188636632558694788, −0.093536226767365081491436004493, 1.12340535259551891111250804177, 2.66915056983477117225727811333, 3.92482333717628935166262903863, 4.83017645484062459915117512520, 6.90805245691473854402183033937, 7.48017788756906224773652467347, 8.338365071770451267997248526752, 9.763608133289875033949926385818, 10.07376172733801163906805780216, 11.14120226293851684863610235885

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