Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.296 + 0.955i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.36 − 1.84i)3-s + (−7.65 − 1.34i)5-s + (10.4 − 8.78i)7-s + (2.18 − 8.73i)9-s + (2.55 − 0.450i)11-s + (−8.23 + 2.99i)13-s + (−20.5 + 10.9i)15-s + (15.2 + 8.82i)17-s + (1.46 + 2.54i)19-s + (8.55 − 40.1i)21-s + (−11.8 + 14.1i)23-s + (33.2 + 12.1i)25-s + (−10.9 − 24.6i)27-s + (1.05 − 2.91i)29-s + (30.6 + 25.7i)31-s + ⋯
L(s)  = 1  + (0.788 − 0.615i)3-s + (−1.53 − 0.269i)5-s + (1.49 − 1.25i)7-s + (0.242 − 0.970i)9-s + (0.232 − 0.0409i)11-s + (−0.633 + 0.230i)13-s + (−1.37 + 0.728i)15-s + (0.899 + 0.519i)17-s + (0.0773 + 0.133i)19-s + (0.407 − 1.91i)21-s + (−0.517 + 0.616i)23-s + (1.33 + 0.484i)25-s + (−0.405 − 0.914i)27-s + (0.0365 − 0.100i)29-s + (0.988 + 0.829i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.296 + 0.955i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (77, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.296 + 0.955i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.22115 - 0.899609i\)
\(L(\frac12)\)  \(\approx\)  \(1.22115 - 0.899609i\)
\(L(2)\)   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 \( 1 \)
3 \( 1 + (-2.36 + 1.84i)T \)
good5 \( 1 + (7.65 + 1.34i)T + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (-10.4 + 8.78i)T + (8.50 - 48.2i)T^{2} \)
11 \( 1 + (-2.55 + 0.450i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (8.23 - 2.99i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (-15.2 - 8.82i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.46 - 2.54i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (11.8 - 14.1i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-1.05 + 2.91i)T + (-644. - 540. i)T^{2} \)
31 \( 1 + (-30.6 - 25.7i)T + (166. + 946. i)T^{2} \)
37 \( 1 + (12.8 - 22.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-21.2 - 58.3i)T + (-1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-5.71 - 32.4i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (33.4 + 39.8i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 53.2iT - 2.80e3T^{2} \)
59 \( 1 + (-102. - 18.0i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (4.56 - 3.82i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (44.4 - 16.1i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (77.5 + 44.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-33.6 - 58.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-34.1 - 12.4i)T + (4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (-6.94 + 19.0i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-83.7 + 48.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (11.6 + 66.2i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−13.35380166623841321889150722040, −12.03008718739786579682735859351, −11.50800899524496988534217284245, −10.07776170155724553975698461138, −8.307607145138798503306511969963, −7.912450748209270560786723771170, −7.03823698968335976728029820317, −4.63478479727764350305382129927, −3.63230856485799713348279019666, −1.23150171543428480990526725699, 2.57832448782964776518950783478, 4.13860763918609623658883782987, 5.22518868344146846839557025412, 7.57754002580059686748993246304, 8.136947596550386865799256412482, 9.129598024006038321038493055714, 10.62125637923530805525962594022, 11.68557822970619108973793568989, 12.25629345037647391852593610250, 14.17096920371993114907823340493

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