Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.916 + 0.399i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.59 + 1.50i)3-s + (−0.298 + 0.355i)5-s + (−10.1 − 3.69i)7-s + (4.47 − 7.80i)9-s + (−10.2 − 12.1i)11-s + (−3.11 + 17.6i)13-s + (0.240 − 1.37i)15-s + (−22.6 − 13.0i)17-s + (1.77 + 3.08i)19-s + (31.9 − 5.66i)21-s + (1.41 + 3.87i)23-s + (4.30 + 24.4i)25-s + (0.107 + 26.9i)27-s + (41.0 − 7.23i)29-s + (−6.62 + 2.41i)31-s + ⋯
L(s)  = 1  + (−0.865 + 0.501i)3-s + (−0.0597 + 0.0711i)5-s + (−1.44 − 0.527i)7-s + (0.497 − 0.867i)9-s + (−0.930 − 1.10i)11-s + (−0.239 + 1.36i)13-s + (0.0160 − 0.0915i)15-s + (−1.33 − 0.769i)17-s + (0.0936 + 0.162i)19-s + (1.51 − 0.269i)21-s + (0.0613 + 0.168i)23-s + (0.172 + 0.976i)25-s + (0.00397 + 0.999i)27-s + (1.41 − 0.249i)29-s + (−0.213 + 0.0777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.916 + 0.399i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (41, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.916 + 0.399i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0161660 - 0.0775143i\)
\(L(\frac12)\)  \(\approx\)  \(0.0161660 - 0.0775143i\)
\(L(2)\)   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 \( 1 + (2.59 - 1.50i)T \)
good5 \( 1 + (0.298 - 0.355i)T + (-4.34 - 24.6i)T^{2} \)
7 \( 1 + (10.1 + 3.69i)T + (37.5 + 31.4i)T^{2} \)
11 \( 1 + (10.2 + 12.1i)T + (-21.0 + 119. i)T^{2} \)
13 \( 1 + (3.11 - 17.6i)T + (-158. - 57.8i)T^{2} \)
17 \( 1 + (22.6 + 13.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.77 - 3.08i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.41 - 3.87i)T + (-405. + 340. i)T^{2} \)
29 \( 1 + (-41.0 + 7.23i)T + (790. - 287. i)T^{2} \)
31 \( 1 + (6.62 - 2.41i)T + (736. - 617. i)T^{2} \)
37 \( 1 + (-4.92 + 8.53i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (42.8 + 7.56i)T + (1.57e3 + 574. i)T^{2} \)
43 \( 1 + (27.2 - 22.8i)T + (321. - 1.82e3i)T^{2} \)
47 \( 1 + (5.51 - 15.1i)T + (-1.69e3 - 1.41e3i)T^{2} \)
53 \( 1 + 75.6iT - 2.80e3T^{2} \)
59 \( 1 + (18.4 - 21.9i)T + (-604. - 3.42e3i)T^{2} \)
61 \( 1 + (55.9 + 20.3i)T + (2.85e3 + 2.39e3i)T^{2} \)
67 \( 1 + (4.03 - 22.8i)T + (-4.21e3 - 1.53e3i)T^{2} \)
71 \( 1 + (32.2 + 18.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (26.0 + 45.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (20.9 + 118. i)T + (-5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (-115. + 20.2i)T + (6.47e3 - 2.35e3i)T^{2} \)
89 \( 1 + (117. - 67.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (72.3 - 60.6i)T + (1.63e3 - 9.26e3i)T^{2} \)
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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

−13.07954005475039662052394693752, −11.78581317304245188432607477947, −10.90991294362527094404818441751, −9.908936710189470886066882439983, −8.971497041150416897748803148909, −7.02518202496796122851034941276, −6.24010207850410365011451107759, −4.74695166338397857418073854713, −3.29365114587685587998615032013, −0.06024137009470937252543005866, 2.61685501277870394284316699327, 4.80626063337548483813666661919, 6.08206546131276329028583862728, 7.01816783412509704557372255311, 8.373039744341382168851266456423, 9.995682289007270564573832217992, 10.59483696923559057700929569597, 12.22086130124939540907874820180, 12.71025654906753897787448002255, 13.41445130425065388342841357906

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