Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.981 + 0.189i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.54 + 3.08i)2-s + (−3.02 − 15.7i)4-s + 3.58·5-s − 16.2i·7-s + (56.1 + 30.6i)8-s + (−9.12 + 11.0i)10-s + 144. i·11-s − 223.·13-s + (50.1 + 41.4i)14-s + (−237. + 95.0i)16-s − 1.80·17-s + 70.1i·19-s + (−10.8 − 56.2i)20-s + (−446. − 368. i)22-s + 251. i·23-s + ⋯
L(s)  = 1  + (−0.636 + 0.771i)2-s + (−0.189 − 0.981i)4-s + 0.143·5-s − 0.331i·7-s + (0.877 + 0.479i)8-s + (−0.0912 + 0.110i)10-s + 1.19i·11-s − 1.32·13-s + (0.255 + 0.211i)14-s + (−0.928 + 0.371i)16-s − 0.00624·17-s + 0.194i·19-s + (−0.0271 − 0.140i)20-s + (−0.922 − 0.761i)22-s + 0.475i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.981 + 0.189i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.981 + 0.189i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.0314945 - 0.330051i\)
\(L(\frac12)\)  \(\approx\)  \(0.0314945 - 0.330051i\)
\(L(3)\)   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 \( 1 + (2.54 - 3.08i)T \)
3 \( 1 \)
good5 \( 1 - 3.58T + 625T^{2} \)
7 \( 1 + 16.2iT - 2.40e3T^{2} \)
11 \( 1 - 144. iT - 1.46e4T^{2} \)
13 \( 1 + 223.T + 2.85e4T^{2} \)
17 \( 1 + 1.80T + 8.35e4T^{2} \)
19 \( 1 - 70.1iT - 1.30e5T^{2} \)
23 \( 1 - 251. iT - 2.79e5T^{2} \)
29 \( 1 + 1.64e3T + 7.07e5T^{2} \)
31 \( 1 + 1.04e3iT - 9.23e5T^{2} \)
37 \( 1 + 690.T + 1.87e6T^{2} \)
41 \( 1 + 993.T + 2.82e6T^{2} \)
43 \( 1 - 1.98e3iT - 3.41e6T^{2} \)
47 \( 1 - 3.19e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.86e3T + 7.89e6T^{2} \)
59 \( 1 - 5.57e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.83e3T + 1.38e7T^{2} \)
67 \( 1 + 5.28e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.87e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.52e3T + 2.83e7T^{2} \)
79 \( 1 - 7.36e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.00e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.14e4T + 6.27e7T^{2} \)
97 \( 1 + 9.02e3T + 8.85e7T^{2} \)
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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

−13.77013328020047740155996662814, −12.58830647569450341845830089412, −11.24072250137281206189410690492, −9.892651550873330731827608497450, −9.462412726198559041523567506605, −7.78043470788728756370121233791, −7.16592608621861003668812007620, −5.70644208594085004220538751470, −4.43977524991144986556668315748, −1.91884358153253504944754350470, 0.17217281424127234926720816060, 2.13564889093580539723554982946, 3.56783471375313475353289362255, 5.31959542051098464161563327452, 7.06379936316866683540687757958, 8.300722213171194149582264641748, 9.267912503939989345148939583029, 10.29033621695366271033463330184, 11.35625267897740550646988217767, 12.21886662162386002302096663926

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