Properties

Degree 1
Conductor 373
Sign $0.118 + 0.992i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.470 + 0.882i)2-s + (0.943 − 0.331i)3-s + (−0.557 + 0.830i)4-s + (0.890 + 0.455i)5-s + (0.736 + 0.676i)6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (0.780 − 0.625i)9-s + (0.0168 + 0.999i)10-s + (0.585 + 0.810i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (0.0843 + 0.996i)14-s + (0.990 + 0.134i)15-s + (−0.378 − 0.925i)16-s + (−0.440 − 0.897i)17-s + ⋯
L(s,χ)  = 1  + (0.470 + 0.882i)2-s + (0.943 − 0.331i)3-s + (−0.557 + 0.830i)4-s + (0.890 + 0.455i)5-s + (0.736 + 0.676i)6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (0.780 − 0.625i)9-s + (0.0168 + 0.999i)10-s + (0.585 + 0.810i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (0.0843 + 0.996i)14-s + (0.990 + 0.134i)15-s + (−0.378 − 0.925i)16-s + (−0.440 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.118 + 0.992i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.118 + 0.992i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $0.118 + 0.992i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (21, \cdot )$
Sato-Tate  :  $\mu(93)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ 0.118 + 0.992i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.916122935 + 1.701470966i$
$L(\frac12,\chi)$  $\approx$  $1.916122935 + 1.701470966i$
$L(\chi,1)$  $\approx$  1.702661413 + 0.9601287115i
$L(1,\chi)$  $\approx$  1.702661413 + 0.9601287115i

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

−24.38710876354367844871124472825, −23.72080493837089080685513145560, −22.02461647841223827569452829705, −21.75416704120690616818158851429, −20.91614742712662876306414730438, −20.16989574545065842923131198284, −19.52157615448383840370134446951, −18.47464823673571992225690853603, −17.44336718047186579634203031299, −16.502502936906653953194758047724, −14.95863384387146712989588012773, −14.3606852363717928555094203540, −13.77817415850671286266208012327, −12.84254796661856271895011519274, −11.88510622112921972926824947527, −10.392997758650436483145356286839, −10.21254677536882821009590416294, −8.76377671487752790417354377076, −8.410623987226530859605223192229, −6.59542360266090134473489195205, −5.25761688242544757752477181338, −4.46275227863912364271176083216, −3.45616350913303561144953173945, −2.13268697870762790980237883974, −1.47354103231897319824848832812, 1.96932581740152191259373897607, 2.69773095075277032176965777264, 4.22707802747625932444480179055, 5.10193643574672245285601919939, 6.44026143295001672310345634463, 7.18644198021271385358115684441, 8.06108005713813647300671802801, 9.218183073766872771633212513544, 9.73239358157953405212263768648, 11.55069460137322718270374498293, 12.51608238542385720660591786844, 13.55028087607695006374735046003, 14.18528248238305959437526514153, 14.90703556350955146076669570099, 15.450348399112179318363965178642, 17.0330089540829473970184783050, 17.78328462529074826011696952724, 18.2771667310260967777500609917, 19.57177006410134249415877205433, 20.64161485805213774153857712366, 21.52739459024296278722694220115, 22.06405523529672763212781717079, 23.203650138849119065185191748497, 24.39827849237048871088180179015, 24.73494259128445485530045364870

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