Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.798 + 0.602i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 − 5.36i)2-s + (−25.5 + 19.2i)4-s + 44.9i·5-s − 28.5i·7-s + (149. + 102. i)8-s + (241. − 80.8i)10-s − 411.·11-s + 684.·13-s + (−152. + 51.2i)14-s + (280. − 984. i)16-s − 1.51e3i·17-s − 2.04e3i·19-s + (−867. − 1.14e3i)20-s + (740. + 2.20e3i)22-s − 3.07e3·23-s + ⋯
L(s)  = 1  + (−0.317 − 0.948i)2-s + (−0.798 + 0.602i)4-s + 0.804i·5-s − 0.219i·7-s + (0.824 + 0.565i)8-s + (0.762 − 0.255i)10-s − 1.02·11-s + 1.12·13-s + (−0.208 + 0.0698i)14-s + (0.273 − 0.961i)16-s − 1.27i·17-s − 1.29i·19-s + (−0.484 − 0.642i)20-s + (0.326 + 0.973i)22-s − 1.21·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.798 + 0.602i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.798 + 0.602i)\)
\(L(3)\)  \(\approx\)  \(0.280219 - 0.836249i\)
\(L(\frac12)\)  \(\approx\)  \(0.280219 - 0.836249i\)
\(L(\frac{7}{2})\)   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 \( 1 + (1.79 + 5.36i)T \)
3 \( 1 \)
good5 \( 1 - 44.9iT - 3.12e3T^{2} \)
7 \( 1 + 28.5iT - 1.68e4T^{2} \)
11 \( 1 + 411.T + 1.61e5T^{2} \)
13 \( 1 - 684.T + 3.71e5T^{2} \)
17 \( 1 + 1.51e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.04e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.07e3T + 6.43e6T^{2} \)
29 \( 1 + 643. iT - 2.05e7T^{2} \)
31 \( 1 - 731. iT - 2.86e7T^{2} \)
37 \( 1 + 5.46e3T + 6.93e7T^{2} \)
41 \( 1 + 6.38e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.79e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.92e4T + 2.29e8T^{2} \)
53 \( 1 + 1.14e4iT - 4.18e8T^{2} \)
59 \( 1 - 424.T + 7.14e8T^{2} \)
61 \( 1 + 6.90e3T + 8.44e8T^{2} \)
67 \( 1 + 5.85e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.91e4T + 1.80e9T^{2} \)
73 \( 1 - 7.02e4T + 2.07e9T^{2} \)
79 \( 1 + 9.52e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.58e4T + 3.93e9T^{2} \)
89 \( 1 - 9.37e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.82e5T + 8.58e9T^{2} \)
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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

−12.14243290180780502205158362023, −11.05133329111448747344240115143, −10.49909238437657364498559508132, −9.299937909399411589177615186052, −8.099241534548488198378182682551, −6.90979203480251488412507668479, −5.09053323832115572049152027649, −3.51283006075854107587642584207, −2.35053125945215032546958628824, −0.39815308574753285575646943263, 1.39859700175376137860115695359, 4.00154275662645942864756118273, 5.40464171286856347182717036018, 6.28256269026533202177792668688, 8.049359217848185677151390483487, 8.424512997165537322579015010747, 9.792623304149944616686863185226, 10.77552608430866133458632237849, 12.48597661085305402151505879297, 13.22163703854405903384849507472

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