Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 23^{2} $
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 + 3-s − 4-s + 2·5-s − 6-s + 7-s + 3·8-s + 9-s − 2·10-s − 4·11-s − 12-s − 2·13-s − 14-s + 2·15-s − 16-s + 6·17-s − 18-s − 4·19-s − 2·20-s + 21-s + 4·22-s + 3·24-s − 25-s + 2·26-s + 27-s − 28-s − 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s + 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11109\)    =    \(3 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11109} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 11109,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.593875657$
$L(\frac12)$  $\approx$  $1.593875657$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.54889524378139, −16.12721077917573, −15.25366721666811, −14.66925374684455, −14.20722956314196, −13.66376007614563, −13.15015001604370, −12.63613257600360, −11.99688166384301, −10.88631993171876, −10.48082909631966, −9.911067176852529, −9.565274802747360, −8.825779665838237, −8.258459326199391, −7.700706042617390, −7.294053011772538, −6.210073091367256, −5.376923206001885, −5.026507186152663, −4.125245764550922, −3.262243590819377, −2.278617602622956, −1.761238006117855, −0.6355519755353798, 0.6355519755353798, 1.761238006117855, 2.278617602622956, 3.262243590819377, 4.125245764550922, 5.026507186152663, 5.376923206001885, 6.210073091367256, 7.294053011772538, 7.700706042617390, 8.258459326199391, 8.825779665838237, 9.565274802747360, 9.911067176852529, 10.48082909631966, 10.88631993171876, 11.99688166384301, 12.63613257600360, 13.15015001604370, 13.66376007614563, 14.20722956314196, 14.66925374684455, 15.25366721666811, 16.12721077917573, 16.54889524378139

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