Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.955 - 0.293i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.419 − 2.79i)2-s + (−7.64 + 2.34i)4-s + 1.49i·5-s + 26.1i·7-s + (9.78 + 20.4i)8-s + (4.18 − 0.627i)10-s + 56.3·11-s − 41.3·13-s + (73.2 − 10.9i)14-s + (52.9 − 35.9i)16-s + 51.0i·17-s + 79.0i·19-s + (−3.51 − 11.4i)20-s + (−23.6 − 157. i)22-s + 27.3·23-s + ⋯
L(s)  = 1  + (−0.148 − 0.988i)2-s + (−0.955 + 0.293i)4-s + 0.133i·5-s + 1.41i·7-s + (0.432 + 0.901i)8-s + (0.132 − 0.0198i)10-s + 1.54·11-s − 0.881·13-s + (1.39 − 0.209i)14-s + (0.827 − 0.561i)16-s + 0.728i·17-s + 0.954i·19-s + (−0.0392 − 0.127i)20-s + (−0.229 − 1.52i)22-s + 0.248·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.955 - 0.293i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.955 - 0.293i)\)
\(L(2)\)  \(\approx\)  \(1.21623 + 0.182582i\)
\(L(\frac12)\)  \(\approx\)  \(1.21623 + 0.182582i\)
\(L(\frac{5}{2})\)   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 \( 1 + (0.419 + 2.79i)T \)
3 \( 1 \)
good5 \( 1 - 1.49iT - 125T^{2} \)
7 \( 1 - 26.1iT - 343T^{2} \)
11 \( 1 - 56.3T + 1.33e3T^{2} \)
13 \( 1 + 41.3T + 2.19e3T^{2} \)
17 \( 1 - 51.0iT - 4.91e3T^{2} \)
19 \( 1 - 79.0iT - 6.85e3T^{2} \)
23 \( 1 - 27.3T + 1.21e4T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 - 187. iT - 2.97e4T^{2} \)
37 \( 1 + 196.T + 5.06e4T^{2} \)
41 \( 1 - 298. iT - 6.89e4T^{2} \)
43 \( 1 - 465. iT - 7.95e4T^{2} \)
47 \( 1 + 373.T + 1.03e5T^{2} \)
53 \( 1 + 620. iT - 1.48e5T^{2} \)
59 \( 1 + 321.T + 2.05e5T^{2} \)
61 \( 1 - 674.T + 2.26e5T^{2} \)
67 \( 1 + 576. iT - 3.00e5T^{2} \)
71 \( 1 - 223.T + 3.57e5T^{2} \)
73 \( 1 - 70.1T + 3.89e5T^{2} \)
79 \( 1 + 1.05e3iT - 4.93e5T^{2} \)
83 \( 1 - 1.21e3T + 5.71e5T^{2} \)
89 \( 1 + 1.34e3iT - 7.04e5T^{2} \)
97 \( 1 + 576.T + 9.12e5T^{2} \)
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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

−12.83491430014718585671156536409, −12.12057151165188782550826648721, −11.41722892828938619077520190901, −10.01617620429081015266280104083, −9.137317096982558428508631519488, −8.221135912219623246630480468533, −6.35114173461415016276057747944, −4.87214463881402183088774002466, −3.28983459568688344859573850367, −1.75961764882611012292001948525, 0.78317034622281289097070280781, 3.90722803729044424184236502567, 5.02229856469833426003730812215, 6.82602137479008418459105861417, 7.23703829806877555794924245999, 8.814815546069073166802732382070, 9.688209856611278888574083000385, 10.87908821912150129780229071487, 12.31897117083439183877109997066, 13.58933647804810167792690105445

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