Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.944 + 0.327i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.61 + 1.08i)2-s + (2.79 + 4.38i)3-s + (5.65 − 5.66i)4-s + (−1.04 + 2.86i)5-s + (−12.0 − 8.43i)6-s + (−28.5 − 5.03i)7-s + (−8.63 + 20.9i)8-s + (−11.4 + 24.4i)9-s + (−0.380 − 8.62i)10-s + (41.0 − 14.9i)11-s + (40.5 + 8.98i)12-s + (−49.8 + 41.8i)13-s + (80.0 − 17.7i)14-s + (−15.4 + 3.42i)15-s + (−0.0980 − 63.9i)16-s + (−54.6 + 31.5i)17-s + ⋯
L(s)  = 1  + (−0.923 + 0.383i)2-s + (0.536 + 0.843i)3-s + (0.706 − 0.707i)4-s + (−0.0934 + 0.256i)5-s + (−0.819 − 0.573i)6-s + (−1.54 − 0.271i)7-s + (−0.381 + 0.924i)8-s + (−0.423 + 0.905i)9-s + (−0.0120 − 0.272i)10-s + (1.12 − 0.409i)11-s + (0.976 + 0.216i)12-s + (−1.06 + 0.893i)13-s + (1.52 − 0.339i)14-s + (−0.266 + 0.0590i)15-s + (−0.00153 − 0.999i)16-s + (−0.779 + 0.449i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.944 + 0.327i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.944 + 0.327i)\)
\(L(2)\)  \(\approx\)  \(0.0660061 - 0.391482i\)
\(L(\frac12)\)  \(\approx\)  \(0.0660061 - 0.391482i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.61 - 1.08i)T \)
3 \( 1 + (-2.79 - 4.38i)T \)
good5 \( 1 + (1.04 - 2.86i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (28.5 + 5.03i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (-41.0 + 14.9i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (49.8 - 41.8i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (54.6 - 31.5i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (112. + 64.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (16.6 + 94.1i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (50.6 - 60.3i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-31.3 + 5.53i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (178. + 309. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-165. - 197. i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-89.4 - 245. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (43.0 - 244. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 - 493. iT - 1.48e5T^{2} \)
59 \( 1 + (-462. - 168. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (65.4 - 371. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (101. + 120. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-157. - 272. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (201. - 348. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (252. - 301. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-156. - 130. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (524. + 302. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.41e3 + 514. i)T + (6.99e5 - 5.86e5i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.25634445603451771829000433306, −12.79359593619113607094031451295, −11.22309990250060687884319711939, −10.35764392912293717120741431960, −9.264754552279938645370026117100, −8.876146149026309788495890311335, −7.10182652914583622603482777924, −6.29413253964636975458268601373, −4.24927487955474807709963690346, −2.61340464741306292945521385806, 0.25362044163853076074424624297, 2.22485164227549033477068747504, 3.56463679717399247662145279450, 6.35781121923033612853880559110, 7.10763685790794392472627471278, 8.441420305051039147887708780959, 9.332451421273338543016098709137, 10.16186719736727175381915794064, 11.91418884885921485758371993762, 12.48636519286564286215206685366

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