Properties

Degree 1
Conductor 11
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s − 21-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s − 29-s − 30-s + ⋯
L(s,χ)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s − 21-s + 23-s − 24-s + 25-s + 26-s + 27-s − 28-s − 29-s − 30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{11} (10, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 11,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9915770035$
$L(\frac12,\chi)$  $\approx$  $0.9915770035$
$L(\chi,1)$  $\approx$  0.9472258250
$L(1,\chi)$  $\approx$  0.9472258250

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−45.03443041544910895447298842202, −44.22866228755402915071954091702, −42.63941507630696800457616302101, −41.49315644722970854417895464060, −38.95426138432898441184537352440, −37.66796277038011510457643735308, −36.65347629447046358323959856101, −35.485085187617277263778295767982, −33.52280264533323492464912935790, −32.10991724081501633365902470335, −29.97469117254131187808345346222, −28.832343001278275591309074781924, −26.78689568412247833195118491511, −25.68596243541730222680937264738, −24.67283686109554841773882275166, −21.638177818276573522470778958792, −20.06759332864609363804035036489, −18.79724653616265280786110493309, −16.99071070103014279771587435499, −15.10915824669017999758236697486, −13.04011532881724693326998338500, −10.10833735739279668002774158265, −8.97128436849938383436174724130, −6.80070840838651795033630280388, −2.47724371122923425905188980494, 2.47724371122923425905188980494, 6.80070840838651795033630280388, 8.97128436849938383436174724130, 10.10833735739279668002774158265, 13.04011532881724693326998338500, 15.10915824669017999758236697486, 16.99071070103014279771587435499, 18.79724653616265280786110493309, 20.06759332864609363804035036489, 21.638177818276573522470778958792, 24.67283686109554841773882275166, 25.68596243541730222680937264738, 26.78689568412247833195118491511, 28.832343001278275591309074781924, 29.97469117254131187808345346222, 32.10991724081501633365902470335, 33.52280264533323492464912935790, 35.485085187617277263778295767982, 36.65347629447046358323959856101, 37.66796277038011510457643735308, 38.95426138432898441184537352440, 41.49315644722970854417895464060, 42.63941507630696800457616302101, 44.22866228755402915071954091702, 45.03443041544910895447298842202

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