Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-0.773 - 0.633i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 − 3.76i)2-s + (−12.3 − 10.1i)4-s + 11.0·5-s − 11.9i·7-s + (−54.8 + 32.9i)8-s + (14.8 − 41.5i)10-s − 219. i·11-s − 37.0·13-s + (−44.8 − 16.0i)14-s + (50.2 + 251. i)16-s + 284.·17-s + 45.4i·19-s + (−136. − 111. i)20-s + (−826. − 295. i)22-s + 201. i·23-s + ⋯
L(s)  = 1  + (0.336 − 0.941i)2-s + (−0.773 − 0.633i)4-s + 0.441·5-s − 0.243i·7-s + (−0.857 + 0.514i)8-s + (0.148 − 0.415i)10-s − 1.81i·11-s − 0.219·13-s + (−0.228 − 0.0818i)14-s + (0.196 + 0.980i)16-s + 0.982·17-s + 0.126i·19-s + (−0.341 − 0.279i)20-s + (−1.70 − 0.610i)22-s + 0.380i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-0.773 - 0.633i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ -0.773 - 0.633i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.134912109\)
\(L(\frac12)\)  \(\approx\)  \(1.134912109\)
\(L(3)\)   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 \( 1 + (-1.34 + 3.76i)T \)
3 \( 1 \)
good5 \( 1 - 11.0T + 625T^{2} \)
7 \( 1 + 11.9iT - 2.40e3T^{2} \)
11 \( 1 + 219. iT - 1.46e4T^{2} \)
13 \( 1 + 37.0T + 2.85e4T^{2} \)
17 \( 1 - 284.T + 8.35e4T^{2} \)
19 \( 1 - 45.4iT - 1.30e5T^{2} \)
23 \( 1 - 201. iT - 2.79e5T^{2} \)
29 \( 1 + 1.22e3T + 7.07e5T^{2} \)
31 \( 1 + 1.51e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.52e3T + 1.87e6T^{2} \)
41 \( 1 + 2.63e3T + 2.82e6T^{2} \)
43 \( 1 - 39.9iT - 3.41e6T^{2} \)
47 \( 1 - 2.88e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.41e3T + 7.89e6T^{2} \)
59 \( 1 - 2.83e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.25e3T + 1.38e7T^{2} \)
67 \( 1 - 930. iT - 2.01e7T^{2} \)
71 \( 1 - 1.16e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.16e3T + 2.83e7T^{2} \)
79 \( 1 + 7.48e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.11e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.73e3T + 6.27e7T^{2} \)
97 \( 1 - 1.20e4T + 8.85e7T^{2} \)
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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

−10.52525167274403374941616186966, −9.672543314607731586504232050269, −8.790190441943074007127688328259, −7.69843572331534110245808102089, −5.97975388293203523435189068294, −5.49121703268298408381470045216, −3.93615069767969569744208019155, −3.05349652666263407840638693655, −1.60549986580785522076104392510, −0.29436721515736894729296114345, 1.91765901170295676693670309664, 3.54534530842812220366022762593, 4.83708187094957442828134728032, 5.56724195906937833610745027135, 6.84228655334752877675393270228, 7.46138180587416314447551240358, 8.629731213366894991502478734341, 9.608999975876136222585413171974, 10.26811376392465617549703278436, 12.01866480387502989299938006013

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