Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.287 - 0.957i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.821 + 5.59i)2-s + (−30.6 − 9.19i)4-s + 8.15i·5-s − 63.0i·7-s + (76.6 − 164. i)8-s + (−45.6 − 6.69i)10-s + 442.·11-s − 76.3·13-s + (352. + 51.7i)14-s + (855. + 563. i)16-s − 643. i·17-s + 2.18e3i·19-s + (74.9 − 250. i)20-s + (−363. + 2.47e3i)22-s + 2.73e3·23-s + ⋯
L(s)  = 1  + (−0.145 + 0.989i)2-s + (−0.957 − 0.287i)4-s + 0.145i·5-s − 0.485i·7-s + (0.423 − 0.906i)8-s + (−0.144 − 0.0211i)10-s + 1.10·11-s − 0.125·13-s + (0.480 + 0.0705i)14-s + (0.835 + 0.550i)16-s − 0.540i·17-s + 1.39i·19-s + (0.0419 − 0.139i)20-s + (−0.160 + 1.09i)22-s + 1.07·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.287 - 0.957i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.287 - 0.957i)\)
\(L(3)\)  \(\approx\)  \(1.29213 + 0.961521i\)
\(L(\frac12)\)  \(\approx\)  \(1.29213 + 0.961521i\)
\(L(\frac{7}{2})\)   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 \( 1 + (0.821 - 5.59i)T \)
3 \( 1 \)
good5 \( 1 - 8.15iT - 3.12e3T^{2} \)
7 \( 1 + 63.0iT - 1.68e4T^{2} \)
11 \( 1 - 442.T + 1.61e5T^{2} \)
13 \( 1 + 76.3T + 3.71e5T^{2} \)
17 \( 1 + 643. iT - 1.41e6T^{2} \)
19 \( 1 - 2.18e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.73e3T + 6.43e6T^{2} \)
29 \( 1 + 1.50e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.50e3iT - 2.86e7T^{2} \)
37 \( 1 + 4.82e3T + 6.93e7T^{2} \)
41 \( 1 - 1.09e4iT - 1.15e8T^{2} \)
43 \( 1 - 8.35e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.39e4T + 2.29e8T^{2} \)
53 \( 1 + 2.28e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.83e4T + 7.14e8T^{2} \)
61 \( 1 - 5.10e3T + 8.44e8T^{2} \)
67 \( 1 + 3.73e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.50e4T + 1.80e9T^{2} \)
73 \( 1 + 6.52e4T + 2.07e9T^{2} \)
79 \( 1 + 7.32e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.52e4T + 3.93e9T^{2} \)
89 \( 1 + 9.20e3iT - 5.58e9T^{2} \)
97 \( 1 - 1.25e5T + 8.58e9T^{2} \)
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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

−13.23624856663854824511337568441, −12.06494512806177720463634117841, −10.62413480777224323126866388338, −9.552607492608964470162051639182, −8.526730000716856583507385684688, −7.27004861749847874748330227031, −6.41970998554280741308526315743, −4.99176224836901845403562433160, −3.65640238071442402869230268226, −1.05500315223204899615386919366, 0.904548895537723653401249760891, 2.49821576032129306700121063929, 3.98801802876030196623544225320, 5.30903094319330509447774640734, 7.01538261720994299705501399016, 8.722312827371494492886571217467, 9.200780054091958455090881057908, 10.58017486562462774037398654384, 11.51331462490891912138989972984, 12.40716168213724077019879671066

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