Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.843 - 0.536i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.89 − 0.903i)2-s + (14.3 + 7.03i)4-s + (−19.5 − 33.8i)5-s + (10.5 + 6.10i)7-s + (−49.6 − 40.4i)8-s + (45.5 + 149. i)10-s + (96.1 + 55.5i)11-s + (−104. − 180. i)13-s + (−35.6 − 33.3i)14-s + (156. + 202. i)16-s − 93.3·17-s − 26.8i·19-s + (−42.5 − 623. i)20-s + (−324. − 303. i)22-s + (−757. + 437. i)23-s + ⋯
L(s)  = 1  + (−0.974 − 0.225i)2-s + (0.898 + 0.439i)4-s + (−0.781 − 1.35i)5-s + (0.215 + 0.124i)7-s + (−0.775 − 0.631i)8-s + (0.455 + 1.49i)10-s + (0.794 + 0.458i)11-s + (−0.618 − 1.07i)13-s + (−0.182 − 0.170i)14-s + (0.613 + 0.790i)16-s − 0.323·17-s − 0.0744i·19-s + (−0.106 − 1.55i)20-s + (−0.670 − 0.626i)22-s + (−1.43 + 0.826i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.843 - 0.536i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.843 - 0.536i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.0361008 + 0.124098i\)
\(L(\frac12)\)  \(\approx\)  \(0.0361008 + 0.124098i\)
\(L(3)\)   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 \( 1 + (3.89 + 0.903i)T \)
3 \( 1 \)
good5 \( 1 + (19.5 + 33.8i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-10.5 - 6.10i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-96.1 - 55.5i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (104. + 180. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 93.3T + 8.35e4T^{2} \)
19 \( 1 + 26.8iT - 1.30e5T^{2} \)
23 \( 1 + (757. - 437. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (650. - 1.12e3i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (593. - 342. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.76e3T + 1.87e6T^{2} \)
41 \( 1 + (-39.0 - 67.6i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.40e3 - 811. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.99e3 + 1.15e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 1.31e3T + 7.89e6T^{2} \)
59 \( 1 + (-4.81e3 + 2.78e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.09e3 - 1.88e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-213. + 123. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 4.60e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.56e3T + 2.83e7T^{2} \)
79 \( 1 + (4.48e3 + 2.59e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (1.62e3 + 936. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.16e3T + 6.27e7T^{2} \)
97 \( 1 + (-2.86e3 + 4.97e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−12.22350976513150722866037093338, −11.42035253069662109185963235540, −10.03371491035340391196562124671, −8.997453504831887266095758598336, −8.174470770367714478777980354494, −7.17391573124649717655069935863, −5.32534395905706302007778311202, −3.74707002850539143902270463495, −1.60380452071821603152274323188, −0.07738221424270010369841998867, 2.21721134200016034140986370172, 3.91556152910638941966599864477, 6.21965818990971349820911788786, 7.07252936679654720135977993884, 8.027765843413599953322438306520, 9.317871371988307559662155787124, 10.44787988852526759189622398101, 11.38265678493800400349270286429, 11.96023900384952111856590615178, 14.12999270371596007412755609190

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