Properties

Degree 1
Conductor 373
Sign $-0.259 - 0.965i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0843 − 0.996i)2-s + (−0.315 − 0.948i)3-s + (−0.985 − 0.168i)4-s + (0.990 + 0.134i)5-s + (−0.972 + 0.234i)6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (−0.801 + 0.598i)9-s + (0.217 − 0.975i)10-s + (0.963 + 0.266i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.890 − 0.455i)14-s + (−0.184 − 0.982i)15-s + (0.943 + 0.331i)16-s + (0.347 + 0.937i)17-s + ⋯
L(s,χ)  = 1  + (0.0843 − 0.996i)2-s + (−0.315 − 0.948i)3-s + (−0.985 − 0.168i)4-s + (0.990 + 0.134i)5-s + (−0.972 + 0.234i)6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (−0.801 + 0.598i)9-s + (0.217 − 0.975i)10-s + (0.963 + 0.266i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.890 − 0.455i)14-s + (−0.184 − 0.982i)15-s + (0.943 + 0.331i)16-s + (0.347 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.259 - 0.965i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.259 - 0.965i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $-0.259 - 0.965i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (29, \cdot )$
Sato-Tate  :  $\mu(93)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ -0.259 - 0.965i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8541555891 - 1.114007365i$
$L(\frac12,\chi)$  $\approx$  $0.8541555891 - 1.114007365i$
$L(\chi,1)$  $\approx$  0.9034994247 - 0.7215138251i
$L(1,\chi)$  $\approx$  0.9034994247 - 0.7215138251i

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

−25.05635550397187759459773245752, −23.94543341431388469037010882492, −23.167699776712699824374921975088, −22.308159633860291304735903345123, −21.42831609792196314581886161524, −20.95576874501357982866721324537, −19.63956621941948944959419997539, −18.28452420527303904506642571568, −17.411741270463732972889629395484, −16.66454561886947669897758275686, −16.45652172376077188387526076566, −14.91631366765873094831883287847, −14.26310724793798970459222432607, −13.76079990089365294463067184745, −12.348987187091553254215155711968, −11.12502262760627730743299427898, −10.04493108597324019272523919, −9.299012359840178761068447438024, −8.53195090100492609271776766983, −6.99165381917055162342613180871, −6.28969775203443570682381496789, −5.08900005198834547514173302202, −4.513641471405941245956540964, −3.35883721783154457490311680463, −1.24098980084386131746512540925, 1.1327407255259777924596047640, 2.09209165668942167913537810441, 2.86170996686798655966455913899, 4.63612917576009306858810111802, 5.67070883681508518308568779208, 6.386355102318741931045649110320, 7.9908131155697547424187627089, 8.86325631059909820172505810086, 9.87357477575047589810926211933, 10.98334583506496969906349814950, 11.710504396652973256880267076682, 12.830541461954396377630827149088, 13.08413701183948776699819397105, 14.453569423125418271458135781713, 14.89132210420782558747136740361, 17.15926201019501433187110624190, 17.39595828845530192174214639854, 18.2651937126355439299303945678, 19.08155930424039756590845602838, 19.8290615200559530762562920066, 20.97476542021560876122991717369, 21.75462052035276628047348160250, 22.4439318251589494926451731372, 23.27109612326960805203005188362, 24.46102557628942209390671221898

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