Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.732 + 0.680i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.716 − 5.14i)3-s + (7.52 + 6.31i)5-s + (24.9 + 9.06i)7-s + (−25.9 + 7.37i)9-s + (17.0 − 14.3i)11-s + (13.5 − 77.1i)13-s + (27.1 − 43.2i)15-s + (32.2 − 55.8i)17-s + (35.2 + 60.9i)19-s + (28.8 − 134. i)21-s + (−112. + 41.1i)23-s + (−4.93 − 27.9i)25-s + (56.5 + 128. i)27-s + (30.3 + 171. i)29-s + (139. − 50.9i)31-s + ⋯
L(s)  = 1  + (−0.137 − 0.990i)3-s + (0.673 + 0.565i)5-s + (1.34 + 0.489i)7-s + (−0.961 + 0.273i)9-s + (0.467 − 0.392i)11-s + (0.290 − 1.64i)13-s + (0.466 − 0.744i)15-s + (0.460 − 0.796i)17-s + (0.425 + 0.736i)19-s + (0.299 − 1.39i)21-s + (−1.02 + 0.372i)23-s + (−0.0394 − 0.223i)25-s + (0.403 + 0.915i)27-s + (0.194 + 1.10i)29-s + (0.810 − 0.295i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.732 + 0.680i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.732 + 0.680i)\)
\(L(2)\)  \(\approx\)  \(1.74795 - 0.686588i\)
\(L(\frac12)\)  \(\approx\)  \(1.74795 - 0.686588i\)
\(L(\frac{5}{2})\)   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 \( 1 + (0.716 + 5.14i)T \)
good5 \( 1 + (-7.52 - 6.31i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (-24.9 - 9.06i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-17.0 + 14.3i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (-13.5 + 77.1i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (-32.2 + 55.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-35.2 - 60.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (112. - 41.1i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (-30.3 - 171. i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (-139. + 50.9i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (-144. + 251. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (76.1 - 431. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (76.4 - 64.1i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (290. + 105. i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 + 533.T + 1.48e5T^{2} \)
59 \( 1 + (-320. - 268. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (87.3 + 31.7i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (80.8 - 458. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (451. - 782. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (86.0 + 149. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (27.8 + 157. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (-103. - 584. i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (818. + 1.41e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (323. - 271. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−13.14984651366261861484867947056, −11.97924834197502493418475151239, −11.22013070202695670341124889376, −10.02095304553009314424305515977, −8.378524484746716303832264102482, −7.69239735415786126357937801831, −6.15551959241430771069310041112, −5.33889770824142821704969672963, −2.86454415741425756950525417354, −1.34745734452814713788850219408, 1.67555873610741487310869684772, 4.14349806502729258315273423347, 4.92876582496595874313953429330, 6.35355142031003024402207770351, 8.130926855850786708099682003300, 9.175373271463635433109380279687, 10.07076925170020603579620048301, 11.26610094277624787318382876508, 11.98067867306582676615077064710, 13.73875142780022206206585008403

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