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  + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 2·7-s − 2·9-s + 2·10-s − 2·12-s − 4·13-s + 4·14-s − 15-s − 4·16-s + 2·17-s − 4·18-s + 2·20-s − 2·21-s − 23-s − 4·25-s − 8·26-s + 5·27-s + 4·28-s − 2·30-s + 7·31-s − 8·32-s + 4·34-s + 2·35-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.577·12-s − 1.10·13-s + 1.06·14-s − 0.258·15-s − 16-s + 0.485·17-s − 0.942·18-s + 0.447·20-s − 0.436·21-s − 0.208·23-s − 4/5·25-s − 1.56·26-s + 0.962·27-s + 0.755·28-s − 0.365·30-s + 1.25·31-s − 1.41·32-s + 0.685·34-s + 0.338·35-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  =  $(2,\ 121,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.759399038$
$L(\frac12)$  $\approx$  $1.759399038$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.25097501976354, −17.79333795903685, −17.37835837820819, −16.20404283761334, −15.00014692230140, −14.31911377690275, −13.62205943269834, −12.33713976074744, −11.81631087306956, −10.76517194615392, −9.366222803316740, −7.787896070436458, −6.238700647892145, −5.411750713566177, −4.445744908548100, −2.630448989358390, 2.630448989358390, 4.445744908548100, 5.411750713566177, 6.238700647892145, 7.787896070436458, 9.366222803316740, 10.76517194615392, 11.81631087306956, 12.33713976074744, 13.62205943269834, 14.31911377690275, 15.00014692230140, 16.20404283761334, 17.37835837820819, 17.79333795903685, 19.25097501976354

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