Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.566 + 0.824i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.44 + 2.43i)2-s + (−3.82 − 7.02i)4-s + (−14.6 + 8.45i)5-s + (3.08 + 1.78i)7-s + (22.6 + 0.866i)8-s + (0.610 − 47.8i)10-s + (25.0 − 43.4i)11-s + (−18.9 − 32.9i)13-s + (−8.80 + 4.93i)14-s + (−34.7 + 53.7i)16-s − 84.3i·17-s − 62.9i·19-s + (115. + 70.6i)20-s + (69.3 + 123. i)22-s + (37.6 + 65.1i)23-s + ⋯
L(s)  = 1  + (−0.511 + 0.859i)2-s + (−0.477 − 0.878i)4-s + (−1.31 + 0.756i)5-s + (0.166 + 0.0963i)7-s + (0.999 + 0.0382i)8-s + (0.0193 − 1.51i)10-s + (0.687 − 1.19i)11-s + (−0.405 − 0.701i)13-s + (−0.168 + 0.0941i)14-s + (−0.543 + 0.839i)16-s − 1.20i·17-s − 0.759i·19-s + (1.29 + 0.789i)20-s + (0.671 + 1.19i)22-s + (0.340 + 0.590i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.566 + 0.824i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.566 + 0.824i)\)
\(L(2)\)  \(\approx\)  \(0.474658 - 0.249675i\)
\(L(\frac12)\)  \(\approx\)  \(0.474658 - 0.249675i\)
\(L(\frac{5}{2})\)   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 \( 1 + (1.44 - 2.43i)T \)
3 \( 1 \)
good5 \( 1 + (14.6 - 8.45i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-3.08 - 1.78i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-25.0 + 43.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (18.9 + 32.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 84.3iT - 4.91e3T^{2} \)
19 \( 1 + 62.9iT - 6.85e3T^{2} \)
23 \( 1 + (-37.6 - 65.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (105. + 60.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-17.2 + 9.97i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 17.7T + 5.06e4T^{2} \)
41 \( 1 + (299. - 172. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (113. + 65.3i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (153. - 265. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 479. iT - 1.48e5T^{2} \)
59 \( 1 + (245. + 425. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (49.9 - 86.4i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-536. + 309. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 254.T + 3.57e5T^{2} \)
73 \( 1 - 100.T + 3.89e5T^{2} \)
79 \( 1 + (856. + 494. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-251. + 436. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.01e3iT - 7.04e5T^{2} \)
97 \( 1 + (503. - 872. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.37329512225632316127872196742, −11.59707907716973251122907305056, −11.08560540858152431068881843666, −9.648289011998771317705576281379, −8.427264016076640333973613524643, −7.53336737365387961450004924765, −6.57945742375748918096743242486, −5.04277420588708608018828167939, −3.38166810561192475358255757468, −0.37068278149258161712504420677, 1.59408566353163872504724116725, 3.83159952568253013993951015699, 4.61821992910703449994642824985, 7.10700104897495741547386225443, 8.148015512766316973238074476216, 9.052475332948067608269922211129, 10.23532243646694516509151609946, 11.46911022253242174786521600032, 12.24065967701762377968642398442, 12.78625039718952176915282430047

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