Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $0.529 + 0.848i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.49 − 1.94i)2-s + (8.46 + 13.5i)4-s − 33.2·5-s + 46.1i·7-s + (−3.25 − 63.9i)8-s + (116. + 64.4i)10-s − 73.4i·11-s − 303.·13-s + (89.5 − 161. i)14-s + (−112. + 229. i)16-s − 182.·17-s + 314. i·19-s + (−281. − 451. i)20-s + (−142. + 256. i)22-s + 335. i·23-s + ⋯
L(s)  = 1  + (−0.874 − 0.485i)2-s + (0.529 + 0.848i)4-s − 1.32·5-s + 0.942i·7-s + (−0.0508 − 0.998i)8-s + (1.16 + 0.644i)10-s − 0.607i·11-s − 1.79·13-s + (0.457 − 0.823i)14-s + (−0.440 + 0.897i)16-s − 0.629·17-s + 0.870i·19-s + (−0.703 − 1.12i)20-s + (−0.294 + 0.530i)22-s + 0.635i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $0.529 + 0.848i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ 0.529 + 0.848i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.4122119729\)
\(L(\frac12)\)  \(\approx\)  \(0.4122119729\)
\(L(3)\)   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 \( 1 + (3.49 + 1.94i)T \)
3 \( 1 \)
good5 \( 1 + 33.2T + 625T^{2} \)
7 \( 1 - 46.1iT - 2.40e3T^{2} \)
11 \( 1 + 73.4iT - 1.46e4T^{2} \)
13 \( 1 + 303.T + 2.85e4T^{2} \)
17 \( 1 + 182.T + 8.35e4T^{2} \)
19 \( 1 - 314. iT - 1.30e5T^{2} \)
23 \( 1 - 335. iT - 2.79e5T^{2} \)
29 \( 1 + 714.T + 7.07e5T^{2} \)
31 \( 1 + 1.13e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.00e3T + 1.87e6T^{2} \)
41 \( 1 - 1.11e3T + 2.82e6T^{2} \)
43 \( 1 - 2.51e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.13e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.05e3T + 7.89e6T^{2} \)
59 \( 1 + 1.01e3iT - 1.21e7T^{2} \)
61 \( 1 - 860.T + 1.38e7T^{2} \)
67 \( 1 + 645. iT - 2.01e7T^{2} \)
71 \( 1 + 9.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.89e3T + 2.83e7T^{2} \)
79 \( 1 + 7.81e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.14e3iT - 4.74e7T^{2} \)
89 \( 1 - 7.65e3T + 6.27e7T^{2} \)
97 \( 1 - 1.27e4T + 8.85e7T^{2} \)
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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

−11.01919242399044605526261009726, −9.738476014939085402936006976188, −9.038480954014735274350588770761, −7.87023536712707625122942260118, −7.55904373951335673870775807247, −6.07782622731393911746389375167, −4.50423224271325134088574957433, −3.29798850280180602935477604976, −2.17722758351970528710760103147, −0.30246595127888126117333755579, 0.59249543149585290383364654294, 2.43008964116090036970515450619, 4.17561862959548000887734240120, 5.07421475103809491120829467370, 7.00261220707818395874036693536, 7.17343639553934499062420788397, 8.117439129406468767393341584567, 9.196699656223582167771946487147, 10.15641667395255916992943235172, 10.96932428876174263366889011431

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