Properties

Degree 1
Conductor 2011
Sign $0.685 + 0.728i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0546 + 0.998i)2-s + (0.990 + 0.134i)3-s + (−0.994 + 0.109i)4-s + (0.832 − 0.554i)5-s + (−0.0796 + 0.996i)6-s + (−0.550 − 0.834i)7-s + (−0.163 − 0.986i)8-s + (0.964 + 0.265i)9-s + (0.599 + 0.800i)10-s + (0.399 + 0.916i)11-s + (−0.999 − 0.0250i)12-s + (−0.968 + 0.250i)13-s + (0.803 − 0.595i)14-s + (0.899 − 0.437i)15-s + (0.976 − 0.217i)16-s + (−0.886 + 0.463i)17-s + ⋯
L(s,χ)  = 1  + (0.0546 + 0.998i)2-s + (0.990 + 0.134i)3-s + (−0.994 + 0.109i)4-s + (0.832 − 0.554i)5-s + (−0.0796 + 0.996i)6-s + (−0.550 − 0.834i)7-s + (−0.163 − 0.986i)8-s + (0.964 + 0.265i)9-s + (0.599 + 0.800i)10-s + (0.399 + 0.916i)11-s + (−0.999 − 0.0250i)12-s + (−0.968 + 0.250i)13-s + (0.803 − 0.595i)14-s + (0.899 − 0.437i)15-s + (0.976 − 0.217i)16-s + (−0.886 + 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.685 + 0.728i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.685 + 0.728i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $0.685 + 0.728i$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (42, \cdot )$
Sato-Tate  :  $\mu(2010)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2011,\ (1:\ ),\ 0.685 + 0.728i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.094443182 + 1.336556316i$
$L(\frac12,\chi)$  $\approx$  $3.094443182 + 1.336556316i$
$L(\chi,1)$  $\approx$  1.427406758 + 0.5730651889i
$L(1,\chi)$  $\approx$  1.427406758 + 0.5730651889i

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

−19.6566215413831864936087055439, −19.015092727042469088544713405783, −18.47166160400633655133199037738, −17.86935721641117028872737778070, −16.981737348562875325430603428256, −15.80494728847768083113967128550, −15.04832120371117254744946352079, −14.26458305935559385799255074180, −13.64958030051599267188917152110, −13.3016571025162475319228483362, −12.121297532333317875777717811376, −11.82138673351669028559656980965, −10.52116676658614611318851618300, −9.88110924500603146810082233724, −9.40801729840167545658035556464, −8.68251502153540012381892878608, −7.90719154048299479899080417092, −6.728446809165444579642748266316, −5.91774169170621296639015073414, −5.04257920946381933628099451558, −3.87475145710331182430286619582, −3.08752005790189323145375295179, −2.512406543965708762521256997583, −1.92410506100812931635260626637, −0.733311065076588986283203016806, 0.66094297025408844662550984721, 1.770672852523140452495055865, 2.763916770140648978057631959879, 3.96168049234008704468659188577, 4.5257656653685072142209873604, 5.21041949079254914185817003447, 6.62211353773422609042953991408, 6.85588729949831859306986387603, 7.7741166095121670366574579477, 8.66481749656327712268663980199, 9.310802245924952592926588347, 9.902274028270133919117548287974, 10.38571663031711677913790565958, 12.221868673591190564153041877640, 12.78857761619836557526571356024, 13.497432866597125881916579381172, 14.14240213028997022904247863463, 14.508153375536084072006634089588, 15.65141411306408217179371018323, 16.00442992913435373782197169767, 16.975470189181518962926574147950, 17.5502750099268206253720579567, 18.07615321378343317634331909789, 19.34457788842291952895750109941, 19.805265515191278636220936433781

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