# Properties

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

# Nearby objects

 previous $$L(s,\chi_{11}(9,·))$$ next $$L(s,\chi_{12}(11,·))$$

## 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