Properties

Degree 2
Conductor $ 3^{2} \cdot 7^{2} \cdot 11 $
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 − 2·5-s − 3·8-s − 2·10-s + 11-s − 6·13-s − 16-s + 2·17-s − 4·19-s + 2·20-s + 22-s − 25-s − 6·26-s + 2·29-s − 8·31-s + 5·32-s + 2·34-s + 6·37-s − 4·38-s + 6·40-s + 10·41-s − 4·43-s − 44-s − 8·47-s − 50-s + 6·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 1.66·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s − 1.17·26-s + 0.371·29-s − 1.43·31-s + 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.648·38-s + 0.948·40-s + 1.56·41-s − 0.609·43-s − 0.150·44-s − 1.16·47-s − 0.141·50-s + 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4851} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4851,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.016526775$
$L(\frac12)$  $\approx$  $1.016526775$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−17.75035564774107, −17.20789571844193, −16.49698818020842, −15.94524646551723, −15.01477909408545, −14.62214697977724, −14.48194530499920, −13.39426841826143, −12.84055706224398, −12.31230800413702, −11.83210193603336, −11.17382155688111, −10.26528213038237, −9.528637923425665, −9.059528650182799, −8.065458500957575, −7.669528800367578, −6.779127928718011, −5.970787722475335, −5.121293241044938, −4.527871681600025, −3.898441050533644, −3.134941335082093, −2.152622476359268, −0.4771137969868273, 0.4771137969868273, 2.152622476359268, 3.134941335082093, 3.898441050533644, 4.527871681600025, 5.121293241044938, 5.970787722475335, 6.779127928718011, 7.669528800367578, 8.065458500957575, 9.059528650182799, 9.528637923425665, 10.26528213038237, 11.17382155688111, 11.83210193603336, 12.31230800413702, 12.84055706224398, 13.39426841826143, 14.48194530499920, 14.62214697977724, 15.01477909408545, 15.94524646551723, 16.49698818020842, 17.20789571844193, 17.75035564774107

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