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  =  \((2,\ 108,\ (\ :5/2),\ 0.798 + 0.602i)\)
\(L(3)\)  \(\approx\)  \(2.02561 - 0.678766i\)
\(L(\frac12)\)  \(\approx\)  \(2.02561 - 0.678766i\)
\(L(\frac{7}{2})\)   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

−12.48933723726008985839410801385, −11.50969413546025715445943575837, −10.78807295893077563596843837628, −9.618646842981136751515849476985, −8.628212689407706876738382674807, −6.85776199606141111977705117138, −5.62233123489381465841566243856, −3.99618859751219216046717326599, −2.86485919256637066185294177443, −1.19193493106104924265191653367, 0.977159774007818430844266135207, 3.65112425602596190716564792220, 4.77741836751192374665007891584, 6.09194478016362046229215925424, 7.14225891203937049382598422233, 8.689013553951013795355504809885, 9.008983803668077801988819592305, 10.78979029314633614129731285841, 12.16878984338083403524884388010, 13.08918765661250597402170773146

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