Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 3·5-s − 2·9-s + 2·12-s + 3·15-s + 4·16-s + 6·20-s − 9·23-s + 4·25-s + 5·27-s − 5·31-s + 4·36-s + 7·37-s + 6·45-s − 12·47-s − 4·48-s − 7·49-s + 6·53-s − 15·59-s − 6·60-s − 8·64-s + 13·67-s + 9·69-s − 3·71-s − 4·75-s − 12·80-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s + 0.577·12-s + 0.774·15-s + 16-s + 1.34·20-s − 1.87·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s + 2/3·36-s + 1.15·37-s + 0.894·45-s − 1.75·47-s − 0.577·48-s − 49-s + 0.824·53-s − 1.95·59-s − 0.774·60-s − 64-s + 1.58·67-s + 1.08·69-s − 0.356·71-s − 0.461·75-s − 1.34·80-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(121\)    =    \(11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{121} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 121,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−19.62369699826356, −18.47939680691510, −17.86158704726754, −16.77484218834199, −16.02369681343710, −14.85409036669708, −14.03863540238532, −12.76419543201372, −11.90069246142033, −11.10044716470992, −9.790262208323322, −8.487357851440445, −7.749714893743499, −6.042146097262260, −4.730703207168041, −3.605773261556393, 0, 3.605773261556393, 4.730703207168041, 6.042146097262260, 7.749714893743499, 8.487357851440445, 9.790262208323322, 11.10044716470992, 11.90069246142033, 12.76419543201372, 14.03863540238532, 14.85409036669708, 16.02369681343710, 16.77484218834199, 17.86158704726754, 18.47939680691510, 19.62369699826356

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