Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.267 - 0.963i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.270 − 1.98i)2-s + (−3.85 + 1.07i)4-s − 7.40·5-s + 9.47i·7-s + (3.16 + 7.34i)8-s + (2.00 + 14.6i)10-s − 6.77i·11-s − 14.4·13-s + (18.7 − 2.55i)14-s + (13.7 − 8.25i)16-s − 17.8·17-s − 5.19i·19-s + (28.5 − 7.92i)20-s + (−13.4 + 1.82i)22-s + 24.9i·23-s + ⋯
L(s)  = 1  + (−0.135 − 0.990i)2-s + (−0.963 + 0.267i)4-s − 1.48·5-s + 1.35i·7-s + (0.395 + 0.918i)8-s + (0.200 + 1.46i)10-s − 0.615i·11-s − 1.10·13-s + (1.34 − 0.182i)14-s + (0.856 − 0.515i)16-s − 1.05·17-s − 0.273i·19-s + (1.42 − 0.396i)20-s + (−0.609 + 0.0831i)22-s + 1.08i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.267 - 0.963i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.267 - 0.963i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.120338 + 0.158317i\)
\(L(\frac12)\)  \(\approx\)  \(0.120338 + 0.158317i\)
\(L(2)\)   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 \( 1 + (0.270 + 1.98i)T \)
3 \( 1 \)
good5 \( 1 + 7.40T + 25T^{2} \)
7 \( 1 - 9.47iT - 49T^{2} \)
11 \( 1 + 6.77iT - 121T^{2} \)
13 \( 1 + 14.4T + 169T^{2} \)
17 \( 1 + 17.8T + 289T^{2} \)
19 \( 1 + 5.19iT - 361T^{2} \)
23 \( 1 - 24.9iT - 529T^{2} \)
29 \( 1 + 29.6T + 841T^{2} \)
31 \( 1 + 17.1iT - 961T^{2} \)
37 \( 1 - 6.41T + 1.36e3T^{2} \)
41 \( 1 + 8.64T + 1.68e3T^{2} \)
43 \( 1 + 50.1iT - 1.84e3T^{2} \)
47 \( 1 - 52.0iT - 2.20e3T^{2} \)
53 \( 1 - 82.6T + 2.80e3T^{2} \)
59 \( 1 - 83.7iT - 3.48e3T^{2} \)
61 \( 1 + 8.41T + 3.72e3T^{2} \)
67 \( 1 - 29.0iT - 4.48e3T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 - 2.16T + 5.32e3T^{2} \)
79 \( 1 - 149. iT - 6.24e3T^{2} \)
83 \( 1 - 13.5iT - 6.88e3T^{2} \)
89 \( 1 + 17.8T + 7.92e3T^{2} \)
97 \( 1 + 138.T + 9.40e3T^{2} \)
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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

−13.44877313212108002196802733450, −12.31079871509416210168286102598, −11.72476357320115904484384615050, −10.99187901405261310237159377195, −9.395941384890138234661744269566, −8.574837322348198668940840750848, −7.48694990637604822106887920300, −5.37641742250104891724606986686, −3.99940642699208796017580529048, −2.57708406636371619572450183894, 0.15036262227245284629706915437, 3.96386930486340945850207757335, 4.73643627113021428583584722420, 6.87707821198327328985652088988, 7.41965267939233516901188910055, 8.417495319166231447217197702458, 9.857146379377837818732311365967, 10.92736028792514873252453918928, 12.28512698911986900601552598046, 13.29180144644114077499245994239

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