Properties

Degree 1
Conductor 11
Sign $0.624 - 0.781i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.624 - 0.781i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.624 - 0.781i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $0.624 - 0.781i$
motivic weight  =  \(0\)
character  :  $\chi_{11} (9, \cdot )$
Sato-Tate  :  $\mu(5)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 11,\ (0:\ ),\ 0.624 - 0.781i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3561278323 - 0.1713091896i$
$L(\frac12,\chi)$  $\approx$  $0.3561278323 - 0.1713091896i$
$L(\chi,1)$  $\approx$  0.5805311136 - 0.2129383511i
$L(1,\chi)$  $\approx$  0.5805311136 - 0.2129383511i

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

−45.75877859475394468803196671809, −44.1807543932131508618010042867, −43.36021389069667485514209185454, −42.30708641995038782073842321800, −39.76404580393091217961458428396, −38.48574617295503178661726067050, −36.92818035598601603325885144168, −35.76575289158548786947394304876, −33.99262875889834913169724447328, −32.83760075474434132336180921300, −31.42427523311059427895530059687, −28.83880063287434417342469151843, −27.22591812668902221184551478255, −26.68452916273036899287058730843, −24.72083578855277620657306332268, −23.13907860578284012046137252354, −20.61265054372333079178400924558, −19.50752880683188820541576949914, −17.0589877129008589583878475262, −15.95701473940678460989492756920, −14.391345209913737768557754830718, −11.08604090734966310677523591806, −9.32576278595562580840034682186, −7.66185762868686205184568530025, −4.62935366251112260134094795549, 2.69600408486917233690482283449, 7.20692647129910428845946072570, 8.704161065911558045022104093621, 11.27579624197347705420756999010, 12.62329617142335150431012578865, 15.169105282641905401132167592826, 17.68807603791082994478424454831, 18.89344173569352104267570432504, 19.9906260243439205467866780351, 22.15714139133311048098743192458, 24.33661634366136980503787359604, 25.85623737581020255318374607184, 27.39714864997876724666794402141, 28.9996374398845296933700159472, 30.50526927558957657974996791575, 31.32845534136625909536474896094, 34.5972204305548442140229585985, 35.103589002333235032242136313328, 36.830721606501950842779862714074, 37.8879189468480404229255113447, 39.44319270094730994698826415607, 41.21327507583455505413149754563, 42.73776464425257506936753077255, 44.166582569894431073202390852504, 45.992857876740975256429899631195

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