Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.985 + 0.172i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.63 + 4.33i)2-s + (−5.51 − 31.5i)4-s − 80.5i·5-s − 216. i·7-s + (156. + 90.8i)8-s + (348. + 293. i)10-s − 153.·11-s − 945.·13-s + (938. + 789. i)14-s + (−963. + 347. i)16-s + 2.18e3i·17-s − 719. i·19-s + (−2.53e3 + 443. i)20-s + (558. − 664. i)22-s + 2.62e3·23-s + ⋯
L(s)  = 1  + (−0.643 + 0.765i)2-s + (−0.172 − 0.985i)4-s − 1.44i·5-s − 1.67i·7-s + (0.864 + 0.501i)8-s + (1.10 + 0.926i)10-s − 0.382·11-s − 1.55·13-s + (1.28 + 1.07i)14-s + (−0.940 + 0.339i)16-s + 1.83i·17-s − 0.457i·19-s + (−1.41 + 0.248i)20-s + (0.245 − 0.292i)22-s + 1.03·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.985 + 0.172i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.985 + 0.172i)\)
\(L(3)\)  \(\approx\)  \(0.0356777 - 0.411282i\)
\(L(\frac12)\)  \(\approx\)  \(0.0356777 - 0.411282i\)
\(L(\frac{7}{2})\)   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 \( 1 + (3.63 - 4.33i)T \)
3 \( 1 \)
good5 \( 1 + 80.5iT - 3.12e3T^{2} \)
7 \( 1 + 216. iT - 1.68e4T^{2} \)
11 \( 1 + 153.T + 1.61e5T^{2} \)
13 \( 1 + 945.T + 3.71e5T^{2} \)
17 \( 1 - 2.18e3iT - 1.41e6T^{2} \)
19 \( 1 + 719. iT - 2.47e6T^{2} \)
23 \( 1 - 2.62e3T + 6.43e6T^{2} \)
29 \( 1 - 155. iT - 2.05e7T^{2} \)
31 \( 1 - 4.90e3iT - 2.86e7T^{2} \)
37 \( 1 - 114.T + 6.93e7T^{2} \)
41 \( 1 + 3.35e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.31e4iT - 1.47e8T^{2} \)
47 \( 1 + 9.40e3T + 2.29e8T^{2} \)
53 \( 1 - 1.96e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.15e4T + 7.14e8T^{2} \)
61 \( 1 - 3.32e4T + 8.44e8T^{2} \)
67 \( 1 + 2.34e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.40e3T + 1.80e9T^{2} \)
73 \( 1 - 4.86e3T + 2.07e9T^{2} \)
79 \( 1 + 1.19e4iT - 3.07e9T^{2} \)
83 \( 1 + 8.39e4T + 3.93e9T^{2} \)
89 \( 1 + 3.68e4iT - 5.58e9T^{2} \)
97 \( 1 + 2.82e4T + 8.58e9T^{2} \)
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

−12.54368284611497369389364180495, −10.78171384759465788259106542295, −9.997372285650643615629759062509, −8.847997766032338867551305697459, −7.84045207372581138650463266953, −6.90665851658602142352984054944, −5.22616822611872068806761953269, −4.34016243668438449354999304534, −1.37995032228404282001179454752, −0.19898474730670339305442047665, 2.46088327117755660438746606687, 2.88028317147964214733992774591, 5.10993432733991362851963680515, 6.83863312342791113844352325068, 7.84015377522568902708708093441, 9.299765642069981197726634241412, 9.955827936579738494919302649448, 11.29958903036581308005134901463, 11.84077654990929566247569398401, 12.92608003650598158639953016540

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