Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.479 - 0.877i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.988 − 3.87i)2-s + (−14.0 − 7.66i)4-s − 20.7·5-s + 5.38i·7-s + (−43.5 + 46.8i)8-s + (−20.5 + 80.4i)10-s + 115. i·11-s + 207.·13-s + (20.8 + 5.32i)14-s + (138. + 215. i)16-s − 383.·17-s + 618. i·19-s + (291. + 159. i)20-s + (448. + 114. i)22-s − 82.9i·23-s + ⋯
L(s)  = 1  + (0.247 − 0.968i)2-s + (−0.877 − 0.479i)4-s − 0.830·5-s + 0.109i·7-s + (−0.681 + 0.732i)8-s + (−0.205 + 0.804i)10-s + 0.955i·11-s + 1.22·13-s + (0.106 + 0.0271i)14-s + (0.540 + 0.841i)16-s − 1.32·17-s + 1.71i·19-s + (0.728 + 0.397i)20-s + (0.925 + 0.236i)22-s − 0.156i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.479 - 0.877i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.479 - 0.877i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.593095 + 0.351981i\)
\(L(\frac12)\)  \(\approx\)  \(0.593095 + 0.351981i\)
\(L(3)\)   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 \( 1 + (-0.988 + 3.87i)T \)
3 \( 1 \)
good5 \( 1 + 20.7T + 625T^{2} \)
7 \( 1 - 5.38iT - 2.40e3T^{2} \)
11 \( 1 - 115. iT - 1.46e4T^{2} \)
13 \( 1 - 207.T + 2.85e4T^{2} \)
17 \( 1 + 383.T + 8.35e4T^{2} \)
19 \( 1 - 618. iT - 1.30e5T^{2} \)
23 \( 1 + 82.9iT - 2.79e5T^{2} \)
29 \( 1 + 201.T + 7.07e5T^{2} \)
31 \( 1 - 196. iT - 9.23e5T^{2} \)
37 \( 1 - 340.T + 1.87e6T^{2} \)
41 \( 1 + 2.79e3T + 2.82e6T^{2} \)
43 \( 1 - 254. iT - 3.41e6T^{2} \)
47 \( 1 - 2.25e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.11e3T + 7.89e6T^{2} \)
59 \( 1 + 1.59e3iT - 1.21e7T^{2} \)
61 \( 1 + 6.08e3T + 1.38e7T^{2} \)
67 \( 1 + 7.38e3iT - 2.01e7T^{2} \)
71 \( 1 - 9.34e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.32e3T + 2.83e7T^{2} \)
79 \( 1 - 5.97e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.00e4iT - 4.74e7T^{2} \)
89 \( 1 - 1.36e4T + 6.27e7T^{2} \)
97 \( 1 - 2.17e3T + 8.85e7T^{2} \)
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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

−12.93084458955580812752452555304, −12.06972077963255669303893393022, −11.19360061402805580676405395863, −10.22037925119824875567009600086, −8.971953249916687759133277316368, −7.913724145986136936049132489944, −6.18293726609568318293296600654, −4.55406106794209802551637908215, −3.55987204461433798083148657217, −1.72864674593953113121492134860, 0.28950389968241938465870342288, 3.41761799747451218949114043043, 4.59148165629686441900472388283, 6.08980664148395274516994555388, 7.14285930456211267197797063165, 8.385604271397851155557630979406, 9.047286533595901091444605620355, 10.89666217146270925333368430209, 11.76224664648092130264049324039, 13.35269266253872786518548778861

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