Properties

Degree 2
Conductor $ 11^{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 + 2·3-s − 4-s + 5-s − 2·6-s + 2·7-s + 3·8-s + 9-s − 10-s − 2·12-s − 13-s − 2·14-s + 2·15-s − 16-s + 5·17-s − 18-s − 6·19-s − 20-s + 4·21-s + 2·23-s + 6·24-s − 4·25-s + 26-s − 4·27-s − 2·28-s − 9·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s − 0.277·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s + 1.21·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s + 0.417·23-s + 1.22·24-s − 4/5·25-s + 0.196·26-s − 0.769·27-s − 0.377·28-s − 1.67·29-s − 0.365·30-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  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 121,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.019794861$
$L(\frac12)$  $\approx$  $1.019794861$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−19.74612303328203, −19.09987704190502, −18.31828140895517, −17.30260577318274, −16.72607487774993, −15.05891242936591, −14.43777919770436, −13.66555884841708, −12.70933619053981, −11.10272305542929, −9.950218896470013, −9.127942331541421, −8.264856530445177, −7.493238108878059, −5.448618747638392, −3.897518759791236, −1.982183666204345, 1.982183666204345, 3.897518759791236, 5.448618747638392, 7.493238108878059, 8.264856530445177, 9.127942331541421, 9.950218896470013, 11.10272305542929, 12.70933619053981, 13.66555884841708, 14.43777919770436, 15.05891242936591, 16.72607487774993, 17.30260577318274, 18.31828140895517, 19.09987704190502, 19.74612303328203

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