Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 13^{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 + 6-s − 2·7-s + 8-s + 9-s + 11-s + 12-s − 2·14-s + 16-s − 6·17-s + 18-s + 4·19-s − 2·21-s + 22-s + 6·23-s + 24-s − 5·25-s + 27-s − 2·28-s + 6·29-s − 8·31-s + 32-s + 33-s − 6·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s + 0.192·27-s − 0.377·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11154\)    =    \(2 \cdot 3 \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11154} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 11154,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.979321090$
$L(\frac12)$  $\approx$  $3.979321090$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.24771044772728, −15.83799282451444, −15.28073866818659, −14.77620709171721, −14.16660867965631, −13.42361730149588, −13.31896997201809, −12.59570115047113, −12.03408866266178, −11.24940288547834, −10.87643352989052, −9.991618163567917, −9.364264427602677, −8.999414622396942, −8.133314938555776, −7.392394534219624, −6.851904130613875, −6.284401844358876, −5.527195428774635, −4.724270793977976, −4.053564837660780, −3.389667918288189, −2.701253881689635, −1.988298572526343, −0.7950279909788347, 0.7950279909788347, 1.988298572526343, 2.701253881689635, 3.389667918288189, 4.053564837660780, 4.724270793977976, 5.527195428774635, 6.284401844358876, 6.851904130613875, 7.392394534219624, 8.133314938555776, 8.999414622396942, 9.364264427602677, 9.991618163567917, 10.87643352989052, 11.24940288547834, 12.03408866266178, 12.59570115047113, 13.31896997201809, 13.42361730149588, 14.16660867965631, 14.77620709171721, 15.28073866818659, 15.83799282451444, 16.24771044772728

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