Properties

Degree 1
Conductor 2011
Sign $-0.783 + 0.621i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.194 + 0.980i)2-s + (−0.279 + 0.960i)3-s + (−0.924 − 0.380i)4-s + (0.788 + 0.615i)5-s + (−0.887 − 0.460i)6-s + (0.770 + 0.637i)7-s + (0.553 − 0.833i)8-s + (−0.844 − 0.536i)9-s + (−0.756 + 0.653i)10-s + (0.931 − 0.363i)11-s + (0.623 − 0.781i)12-s + (0.0515 − 0.998i)13-s + (−0.774 + 0.632i)14-s + (−0.810 + 0.585i)15-s + (0.709 + 0.704i)16-s + (−0.701 + 0.713i)17-s + ⋯
L(s,χ)  = 1  + (−0.194 + 0.980i)2-s + (−0.279 + 0.960i)3-s + (−0.924 − 0.380i)4-s + (0.788 + 0.615i)5-s + (−0.887 − 0.460i)6-s + (0.770 + 0.637i)7-s + (0.553 − 0.833i)8-s + (−0.844 − 0.536i)9-s + (−0.756 + 0.653i)10-s + (0.931 − 0.363i)11-s + (0.623 − 0.781i)12-s + (0.0515 − 0.998i)13-s + (−0.774 + 0.632i)14-s + (−0.810 + 0.585i)15-s + (0.709 + 0.704i)16-s + (−0.701 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.783 + 0.621i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.783 + 0.621i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $-0.783 + 0.621i$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (102, \cdot )$
Sato-Tate  :  $\mu(2010)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2011,\ (1:\ ),\ -0.783 + 0.621i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7980067648 + 2.288177692i$
$L(\frac12,\chi)$  $\approx$  $0.7980067648 + 2.288177692i$
$L(\chi,1)$  $\approx$  0.7516551039 + 0.8413893047i
$L(1,\chi)$  $\approx$  0.7516551039 + 0.8413893047i

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

−19.67074206848149681671058586565, −18.621935112394355928725422281904, −18.08290174983621221438929448512, −17.39233909255625732039866592342, −17.01889823164713386865627785861, −16.275072627819612066955510457879, −14.48550238937861730296864925474, −14.163374098187882223237339166677, −13.31026586665949179620108420956, −12.99164545272335064842030057044, −11.83249709003988636642864266140, −11.5511815770274654248937815707, −10.79722150929439894286629696491, −9.7286037969105439029791164918, −8.96195670596412883614021073786, −8.56463437249320195221176940198, −7.19204868392371322489029756035, −6.935412543989827878247867791819, −5.49643762893751647043108436559, −4.8757431299654896951982188579, −4.11563364551570686470311698939, −2.81101509939438846569313470805, −1.8711674065312574613273107269, −1.3463843002696838734111294686, −0.69179665845607860557847036594, 0.73608575379942466339614105685, 1.93509764777937565467788262963, 3.23045704512382911416799999799, 4.02792294273528928223084723611, 5.06788908161030431971958919762, 5.683429824583405990483133649842, 6.14028905294576904718164568648, 7.0931278898065423965463098038, 8.228493066248169045982250770577, 8.783982967900243368002811308772, 9.593949330307658784648522189382, 10.163167893277139906924340138976, 11.033884304556805016262182074132, 11.66180643254717978900797439595, 12.9403880668489461954008391093, 13.738908198945819519095286975, 14.56338985100652130132998037295, 15.10209436725527621837470975581, 15.33830097594360295642143417080, 16.641480403556197571694620235399, 17.01564651791804598246698312308, 17.70062835644705950930448364291, 18.300403320238287047198123856292, 19.02053170788757150485179588347, 20.14808265683144428222834482103

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