Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.997 + 0.0640i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.65 + 0.181i)2-s + (31.9 − 2.04i)4-s − 69.1i·5-s − 238. i·7-s + (−180. + 17.3i)8-s + (12.5 + 391. i)10-s − 350.·11-s + 669.·13-s + (43.2 + 1.35e3i)14-s + (1.01e3 − 130. i)16-s − 1.39e3i·17-s + 1.78e3i·19-s + (−141. − 2.20e3i)20-s + (1.98e3 − 63.5i)22-s − 1.32e3·23-s + ⋯
L(s)  = 1  + (−0.999 + 0.0320i)2-s + (0.997 − 0.0640i)4-s − 1.23i·5-s − 1.84i·7-s + (−0.995 + 0.0959i)8-s + (0.0396 + 1.23i)10-s − 0.874·11-s + 1.09·13-s + (0.0589 + 1.84i)14-s + (0.991 − 0.127i)16-s − 1.17i·17-s + 1.13i·19-s + (−0.0792 − 1.23i)20-s + (0.874 − 0.0280i)22-s − 0.521·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0640i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.997 + 0.0640i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.997 + 0.0640i)\)
\(L(3)\)  \(\approx\)  \(0.0230169 - 0.718514i\)
\(L(\frac12)\)  \(\approx\)  \(0.0230169 - 0.718514i\)
\(L(\frac{7}{2})\)   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 \( 1 + (5.65 - 0.181i)T \)
3 \( 1 \)
good5 \( 1 + 69.1iT - 3.12e3T^{2} \)
7 \( 1 + 238. iT - 1.68e4T^{2} \)
11 \( 1 + 350.T + 1.61e5T^{2} \)
13 \( 1 - 669.T + 3.71e5T^{2} \)
17 \( 1 + 1.39e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.78e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.32e3T + 6.43e6T^{2} \)
29 \( 1 - 6.89e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.33e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.29e4T + 6.93e7T^{2} \)
41 \( 1 + 8.71e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.06e4iT - 1.47e8T^{2} \)
47 \( 1 - 3.98e3T + 2.29e8T^{2} \)
53 \( 1 - 4.05e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.51e4T + 7.14e8T^{2} \)
61 \( 1 - 8.96e3T + 8.44e8T^{2} \)
67 \( 1 + 1.24e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.50e3T + 1.80e9T^{2} \)
73 \( 1 + 4.58e4T + 2.07e9T^{2} \)
79 \( 1 - 3.74e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.86e4T + 3.93e9T^{2} \)
89 \( 1 - 7.93e3iT - 5.58e9T^{2} \)
97 \( 1 - 5.65e4T + 8.58e9T^{2} \)
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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

−12.14781188372673535969264176690, −10.81331794570301401392656606296, −10.16430609285008301630644164663, −8.897895874995991701722827756470, −7.942793212232242598979908874586, −6.98998066886707370416876439210, −5.32748157944248904147846424371, −3.71069326688308417993925828118, −1.40239142103466940700385007682, −0.38632454356948183721764351549, 2.11534485428649399393689800776, 3.07707879279604105124227499357, 5.76741674102834234786256319458, 6.56776392603027890980293987645, 8.060718194596370991810999215758, 8.854689352010842991501450313557, 10.12908096096134825665155155488, 11.03997184380698741123743577871, 11.84407087428610364479036636587, 13.08896670256640312863443142704

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