Properties

Degree 1
Conductor 11
Sign $0.957 - 0.288i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + 12-s + (0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + ⋯
L(s,χ)  = 1  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + 12-s + (0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.957 - 0.288i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.957 - 0.288i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $0.957 - 0.288i$
motivic weight  =  \(0\)
character  :  $\chi_{11} (6, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 11,\ (1:\ ),\ 0.957 - 0.288i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.503380081 - 0.2211997697i$
$L(\frac12,\chi)$  $\approx$  $1.503380081 - 0.2211997697i$
$L(\chi,1)$  $\approx$  1.415747663 - 0.1740328837i
$L(1,\chi)$  $\approx$  1.415747663 - 0.1740328837i

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

−45.571883708328470662009967870960, −43.13903733098448016886778132619, −42.601218189354830328691114239797, −41.34346950295061398221815956035, −39.91816544484692915524132036871, −38.421569250426374648790601150719, −36.103221852765725499410258250258, −35.0748181390130967490481952242, −33.60918441512861464269747264056, −31.76704292714201836153420720265, −30.615315398535949332903421061066, −29.55920151127941976080946946763, −26.67028496809013083238174485224, −25.38976369113433935050441838210, −23.62369920986579610417564305629, −22.95929393378876961523787194156, −20.50797618286869986017769960585, −18.7495375665351457827169284855, −16.64292363517938135933695505995, −14.674032161018154187712356107968, −13.33022743788218395645836666989, −11.59406426254877199887647483811, −7.88643465920150753487622877767, −6.63073045048494212311693469779, −3.54704109171945007666447637176, 3.41492187932220368062022478551, 5.27308657584208989337435824912, 8.95354546437232036116473143789, 11.00919009252031218454364175306, 12.73163403755865549814505940274, 14.94827047321579871059739778452, 16.003353712811180942448748166867, 19.23890660918179306832270433212, 20.5790557747237967713122828270, 21.88489757248339392930656813056, 23.38804121013922687721723923210, 25.294856636069449525514142678791, 27.61231692090569895478179228700, 28.4508199966808353191563583286, 30.758854343797428801601385292783, 31.87403308116394856064712507976, 32.799985112149809068419874207202, 34.72345992628301281730177232071, 37.01755954390176887573149124687, 38.395494452428109122979398955656, 39.22705738819166212922797709288, 40.66058066592356276517670187998, 42.45917275651500585448292842681, 43.63192017509203526976799880310, 45.30426327898638018804634367866

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