Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.918 + 0.394i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.82 + 0.218i)2-s + (−2.56 + 4.51i)3-s + (7.90 − 1.23i)4-s + (2.35 − 6.47i)5-s + (6.24 − 13.3i)6-s + (−6.19 − 1.09i)7-s + (−22.0 + 5.19i)8-s + (−13.8 − 23.1i)9-s + (−5.23 + 18.7i)10-s + (−20.4 + 7.42i)11-s + (−14.7 + 38.8i)12-s + (59.5 − 49.9i)13-s + (17.7 + 1.73i)14-s + (23.2 + 27.2i)15-s + (60.9 − 19.4i)16-s + (34.1 − 19.6i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0771i)2-s + (−0.493 + 0.869i)3-s + (0.988 − 0.153i)4-s + (0.210 − 0.578i)5-s + (0.425 − 0.905i)6-s + (−0.334 − 0.0590i)7-s + (−0.973 + 0.229i)8-s + (−0.512 − 0.858i)9-s + (−0.165 + 0.593i)10-s + (−0.559 + 0.203i)11-s + (−0.354 + 0.935i)12-s + (1.27 − 1.06i)13-s + (0.338 + 0.0330i)14-s + (0.399 + 0.469i)15-s + (0.952 − 0.303i)16-s + (0.486 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.918 + 0.394i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.918 + 0.394i)\)
\(L(2)\)  \(\approx\)  \(0.786139 - 0.161651i\)
\(L(\frac12)\)  \(\approx\)  \(0.786139 - 0.161651i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.82 - 0.218i)T \)
3 \( 1 + (2.56 - 4.51i)T \)
good5 \( 1 + (-2.35 + 6.47i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (6.19 + 1.09i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (20.4 - 7.42i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (-59.5 + 49.9i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-34.1 + 19.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-102. - 59.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (29.8 + 169. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-13.5 + 16.1i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-123. + 21.7i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (100. + 173. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-154. - 184. i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (20.3 + 55.7i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-45.4 + 257. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 - 26.8iT - 1.48e5T^{2} \)
59 \( 1 + (-104. - 37.8i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (-105. + 597. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (506. + 603. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (76.4 + 132. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (273. - 472. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-103. + 123. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (889. + 746. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (-1.20e3 - 697. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.25e3 + 457. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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

−12.88678133455330685546097833205, −11.85000289930723061856562401985, −10.64925514455822015277108945698, −10.03420418726656267725450719131, −8.967652927991033934228063266177, −7.943042152431423214924170883531, −6.25222936390735685630446264400, −5.25840907249757994343375636924, −3.25745233477179283253796812936, −0.74788959590032162780477521999, 1.30991853260479264456107591518, 2.98569531199364680800807409872, 5.78421413075877792236429290775, 6.72309994830468907699630903201, 7.68835751802056164569857190196, 8.907699320437728468003975390140, 10.18691208213082763251402559209, 11.22498743190534129040108323159, 11.86802863122121136464732094557, 13.23008177129490898198447078124

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