Properties

Degree $2$
Conductor $109$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 3·5-s + 2·7-s − 3·8-s − 3·9-s + 3·10-s + 11-s + 2·14-s − 16-s − 8·17-s − 3·18-s − 5·19-s − 3·20-s + 22-s + 7·23-s + 4·25-s − 2·28-s − 5·29-s + 6·31-s + 5·32-s − 8·34-s + 6·35-s + 3·36-s + 2·37-s − 5·38-s − 9·40-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.34·5-s + 0.755·7-s − 1.06·8-s − 9-s + 0.948·10-s + 0.301·11-s + 0.534·14-s − 1/4·16-s − 1.94·17-s − 0.707·18-s − 1.14·19-s − 0.670·20-s + 0.213·22-s + 1.45·23-s + 4/5·25-s − 0.377·28-s − 0.928·29-s + 1.07·31-s + 0.883·32-s − 1.37·34-s + 1.01·35-s + 1/2·36-s + 0.328·37-s − 0.811·38-s − 1.42·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(109\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{109} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 109,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.411025916\)
\(L(\frac12)\) \(\approx\) \(1.411025916\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.43116512905609, −18.16484836937616, −17.45899129495912, −17.03448147474551, −15.10848800648600, −14.57891176592136, −13.51696514179329, −13.16351918145632, −11.68350949166309, −10.66560530782850, −9.150313873860532, −8.647170498136539, −6.520964333213792, −5.536022054609493, −4.449402452525226, −2.448523338630769, 2.448523338630769, 4.449402452525226, 5.536022054609493, 6.520964333213792, 8.647170498136539, 9.150313873860532, 10.66560530782850, 11.68350949166309, 13.16351918145632, 13.51696514179329, 14.57891176592136, 15.10848800648600, 17.03448147474551, 17.45899129495912, 18.16484836937616, 19.43116512905609

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