Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 1

Learn more about

Dirichlet series

 L(s)  = 1 + 57.6·5-s − 188.·7-s − 704.·11-s + 795.·13-s + 180.·17-s − 661.·19-s − 3.63e3·23-s + 196·25-s − 8.02e3·29-s − 3.00e3·31-s − 1.08e4·35-s − 1.52e3·37-s − 3.46e3·41-s + 1.18e4·43-s + 5.97e3·47-s + 1.88e4·49-s − 1.38e4·53-s − 4.05e4·55-s − 2.26e4·59-s − 3.74e4·61-s + 4.58e4·65-s + 7.09e4·67-s + 6.77e4·71-s − 3.15e4·73-s + 1.32e5·77-s + 6.27e4·79-s − 9.34e4·83-s + ⋯
 L(s)  = 1 + 1.03·5-s − 1.45·7-s − 1.75·11-s + 1.30·13-s + 0.151·17-s − 0.420·19-s − 1.43·23-s + 0.0627·25-s − 1.77·29-s − 0.561·31-s − 1.50·35-s − 0.182·37-s − 0.321·41-s + 0.974·43-s + 0.394·47-s + 1.12·49-s − 0.675·53-s − 1.80·55-s − 0.846·59-s − 1.28·61-s + 1.34·65-s + 1.93·67-s + 1.59·71-s − 0.693·73-s + 2.55·77-s + 1.13·79-s − 1.48·83-s + ⋯

Functional equation

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

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-1$ motivic weight = $$5$$ character : $\chi_{108} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 108,\ (\ :5/2),\ -1)$$ $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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$$
3 $$1$$
good5 $$1 - 57.6T + 3.12e3T^{2}$$
7 $$1 + 188.T + 1.68e4T^{2}$$
11 $$1 + 704.T + 1.61e5T^{2}$$
13 $$1 - 795.T + 3.71e5T^{2}$$
17 $$1 - 180.T + 1.41e6T^{2}$$
19 $$1 + 661.T + 2.47e6T^{2}$$
23 $$1 + 3.63e3T + 6.43e6T^{2}$$
29 $$1 + 8.02e3T + 2.05e7T^{2}$$
31 $$1 + 3.00e3T + 2.86e7T^{2}$$
37 $$1 + 1.52e3T + 6.93e7T^{2}$$
41 $$1 + 3.46e3T + 1.15e8T^{2}$$
43 $$1 - 1.18e4T + 1.47e8T^{2}$$
47 $$1 - 5.97e3T + 2.29e8T^{2}$$
53 $$1 + 1.38e4T + 4.18e8T^{2}$$
59 $$1 + 2.26e4T + 7.14e8T^{2}$$
61 $$1 + 3.74e4T + 8.44e8T^{2}$$
67 $$1 - 7.09e4T + 1.35e9T^{2}$$
71 $$1 - 6.77e4T + 1.80e9T^{2}$$
73 $$1 + 3.15e4T + 2.07e9T^{2}$$
79 $$1 - 6.27e4T + 3.07e9T^{2}$$
83 $$1 + 9.34e4T + 3.93e9T^{2}$$
89 $$1 - 6.87e4T + 5.58e9T^{2}$$
97 $$1 - 1.17e5T + 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

−12.66321745753485678525558099401, −10.89412472095523456153149960272, −10.07402872452429506893550481858, −9.199154689048109096497305727703, −7.79586544670057594365706128324, −6.25936863139840133919971146252, −5.59754178581492614688608002415, −3.56943105237776466554535272994, −2.15948891911118990100055856966, 0, 2.15948891911118990100055856966, 3.56943105237776466554535272994, 5.59754178581492614688608002415, 6.25936863139840133919971146252, 7.79586544670057594365706128324, 9.199154689048109096497305727703, 10.07402872452429506893550481858, 10.89412472095523456153149960272, 12.66321745753485678525558099401