Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 19^{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 + 4·7-s + 3·8-s + 9-s + 2·10-s + 11-s − 12-s + 2·13-s − 4·14-s − 2·15-s − 16-s − 2·17-s − 18-s + 2·20-s + 4·21-s − 22-s + 8·23-s + 3·24-s − 25-s − 2·26-s + 27-s − 4·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.447·20-s + 0.872·21-s − 0.213·22-s + 1.66·23-s + 0.612·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11913 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11913 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11913\)    =    \(3 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11913} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 11913,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.888986227$
$L(\frac12)$  $\approx$  $1.888986227$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 - T \)
19 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.40201619923625, −15.71786831056715, −15.31417438010237, −14.69685334423085, −14.10311022796278, −13.71344377506442, −13.09366315738481, −12.28428839084235, −11.66861537751474, −11.11217957285425, −10.62908674506778, −9.972868756108570, −9.098114636733542, −8.587231688159157, −8.392753525368326, −7.678045574293424, −7.249649671902821, −6.407907497873899, −5.193876157476620, −4.685821836404851, −4.184878624534243, −3.439242498411180, −2.393012063704497, −1.382888141373656, −0.7869914007676585, 0.7869914007676585, 1.382888141373656, 2.393012063704497, 3.439242498411180, 4.184878624534243, 4.685821836404851, 5.193876157476620, 6.407907497873899, 7.249649671902821, 7.678045574293424, 8.392753525368326, 8.587231688159157, 9.098114636733542, 9.972868756108570, 10.62908674506778, 11.11217957285425, 11.66861537751474, 12.28428839084235, 13.09366315738481, 13.71344377506442, 14.10311022796278, 14.69685334423085, 15.31417438010237, 15.71786831056715, 16.40201619923625

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