Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.875 - 0.483i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 + 0.947i)2-s + (5.14 − 0.697i)3-s + (6.20 − 5.04i)4-s + (−1.80 + 4.95i)5-s + (−13.0 + 6.73i)6-s + (14.7 + 2.60i)7-s + (−11.7 + 19.3i)8-s + (26.0 − 7.18i)9-s + (0.112 − 14.8i)10-s + (−10.3 + 3.77i)11-s + (28.4 − 30.3i)12-s + (13.1 − 11.0i)13-s + (−41.8 + 7.05i)14-s + (−5.82 + 26.7i)15-s + (13.0 − 62.6i)16-s + (48.2 − 27.8i)17-s + ⋯
L(s)  = 1  + (−0.942 + 0.334i)2-s + (0.990 − 0.134i)3-s + (0.775 − 0.631i)4-s + (−0.161 + 0.442i)5-s + (−0.888 + 0.458i)6-s + (0.798 + 0.140i)7-s + (−0.519 + 0.854i)8-s + (0.963 − 0.265i)9-s + (0.00355 − 0.471i)10-s + (−0.284 + 0.103i)11-s + (0.683 − 0.729i)12-s + (0.280 − 0.235i)13-s + (−0.799 + 0.134i)14-s + (−0.100 + 0.460i)15-s + (0.203 − 0.979i)16-s + (0.688 − 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.875 - 0.483i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.875 - 0.483i)\)
\(L(2)\)  \(\approx\)  \(1.51522 + 0.391029i\)
\(L(\frac12)\)  \(\approx\)  \(1.51522 + 0.391029i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.66 - 0.947i)T \)
3 \( 1 + (-5.14 + 0.697i)T \)
good5 \( 1 + (1.80 - 4.95i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (-14.7 - 2.60i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (10.3 - 3.77i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (-13.1 + 11.0i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-48.2 + 27.8i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-43.9 - 25.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-33.4 - 189. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-19.0 + 22.7i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-42.3 + 7.46i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (161. + 278. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (255. + 304. i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (24.4 + 67.1i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (59.0 - 334. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + 357. iT - 1.48e5T^{2} \)
59 \( 1 + (136. + 49.7i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (125. - 713. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (193. + 230. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (384. + 665. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-213. + 370. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (506. - 604. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (477. + 400. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (757. + 437. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.22e3 - 447. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−13.64975545427207903411533091795, −12.06831044016583884367384131234, −10.95691716174528446651744566272, −9.873166491813142712771340869620, −8.861744076929708681212973356888, −7.79786422130222342036960129723, −7.17261829826274636373313350978, −5.40158169533720723344181200105, −3.19538018386223267256204958710, −1.56550891982619032030909917266, 1.34314496053324240292602325902, 2.98088566583292558183753416237, 4.59656308521643645949486106238, 6.85690617404305865035527260963, 8.208458446960871172919956026970, 8.524812610959674195874304426004, 9.854769288680911922978555530346, 10.75960516298878505983697793774, 12.01212149734596052669765503950, 13.01872943571216301342528201531

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