Properties

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

Origins

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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

\( d \)  =  \(2\)
\( N \)  =  \(109\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{109} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 109,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.411025916$
$L(\frac12)$  $\approx$  $1.411025916$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 109$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 109$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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 \( 1 - T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−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