Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.293 + 0.956i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.40 + 2.76i)3-s + (−14.9 + 12.5i)5-s + (23.5 − 8.56i)7-s + (11.7 − 24.3i)9-s + (−39.5 − 33.1i)11-s + (−10.2 − 57.8i)13-s + (31.1 − 96.4i)15-s + (−3.52 − 6.09i)17-s + (−42.6 + 73.9i)19-s + (−79.9 + 102. i)21-s + (32.2 + 11.7i)23-s + (44.3 − 251. i)25-s + (15.3 + 139. i)27-s + (37.0 − 210. i)29-s + (−133. − 48.7i)31-s + ⋯
L(s)  = 1  + (−0.847 + 0.531i)3-s + (−1.33 + 1.12i)5-s + (1.27 − 0.462i)7-s + (0.435 − 0.900i)9-s + (−1.08 − 0.909i)11-s + (−0.217 − 1.23i)13-s + (0.536 − 1.65i)15-s + (−0.0502 − 0.0869i)17-s + (−0.515 + 0.892i)19-s + (−0.830 + 1.06i)21-s + (0.292 + 0.106i)23-s + (0.354 − 2.01i)25-s + (0.109 + 0.994i)27-s + (0.237 − 1.34i)29-s + (−0.776 − 0.282i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.293 + 0.956i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.293 + 0.956i)\)
\(L(2)\)  \(\approx\)  \(0.201061 - 0.271989i\)
\(L(\frac12)\)  \(\approx\)  \(0.201061 - 0.271989i\)
\(L(\frac{5}{2})\)   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 \( 1 + (4.40 - 2.76i)T \)
good5 \( 1 + (14.9 - 12.5i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (-23.5 + 8.56i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (39.5 + 33.1i)T + (231. + 1.31e3i)T^{2} \)
13 \( 1 + (10.2 + 57.8i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (3.52 + 6.09i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (42.6 - 73.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-32.2 - 11.7i)T + (9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (-37.0 + 210. i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (133. + 48.7i)T + (2.28e4 + 1.91e4i)T^{2} \)
37 \( 1 + (65.2 + 113. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (8.52 + 48.3i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (-126. - 106. i)T + (1.38e4 + 7.82e4i)T^{2} \)
47 \( 1 + (470. - 171. i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 + 347.T + 1.48e5T^{2} \)
59 \( 1 + (172. - 144. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (577. - 210. i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (162. + 919. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (-46.7 - 80.9i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (133. - 232. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (155. - 883. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (-6.17 + 35.0i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (-361. + 626. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.21e3 + 1.01e3i)T + (1.58e5 + 8.98e5i)T^{2} \)
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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

−12.65248703826166854876512505368, −11.42650947687863173575471169480, −10.94107783972579183886987012040, −10.28872365240665014750362486971, −8.110922170191795397250931564281, −7.55165528237014920849819669946, −5.91823479305950079254186324925, −4.57575146828871632670397510626, −3.29713842409239977063966064590, −0.20897524598655390502915215666, 1.70050061846561073016380161155, 4.70041630879142420883421106461, 4.96392633940852462464981905643, 7.05775200533576083738834963991, 7.956255124692723444130905566512, 8.916878359396615552204700350762, 10.82042487463034888577268861420, 11.62494242979697473732757336076, 12.32618987723418235138941281434, 13.10328360867929113520518748791

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