Properties

Degree 2
Conductor $ 3^{2} \cdot 7^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

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 + 2·13-s − 16-s − 2·17-s + 2·20-s + 22-s − 8·23-s − 25-s − 2·26-s + 6·29-s + 8·31-s − 5·32-s + 2·34-s + 6·37-s − 6·40-s − 2·41-s + 44-s + 8·46-s + 8·47-s + 50-s − 2·52-s − 6·53-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 + 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.213·22-s − 1.66·23-s − 1/5·25-s − 0.392·26-s + 1.11·29-s + 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.948·40-s − 0.312·41-s + 0.150·44-s + 1.17·46-s + 1.16·47-s + 0.141·50-s − 0.277·52-s − 0.824·53-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  =  1
Selberg data  =  $(2,\ 4851,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
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

−18.17703892648963, −17.61139167093959, −16.96897428321663, −16.27717092249391, −15.67578187065813, −15.40262663240886, −14.32120031973275, −13.78448974184733, −13.35221367700233, −12.38564994751500, −11.95106138211592, −11.14738345986964, −10.53417019561992, −9.924673928821695, −9.269139072903253, −8.402747192524386, −8.092649848612793, −7.551136104271194, −6.551187365777603, −5.832782961732948, −4.659706463964552, −4.284999257340347, −3.436322805351164, −2.267521067680484, −1.033269222569597, 0, 1.033269222569597, 2.267521067680484, 3.436322805351164, 4.284999257340347, 4.659706463964552, 5.832782961732948, 6.551187365777603, 7.551136104271194, 8.092649848612793, 8.402747192524386, 9.269139072903253, 9.924673928821695, 10.53417019561992, 11.14738345986964, 11.95106138211592, 12.38564994751500, 13.35221367700233, 13.78448974184733, 14.32120031973275, 15.40262663240886, 15.67578187065813, 16.27717092249391, 16.96897428321663, 17.61139167093959, 18.17703892648963

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