Properties

Degree $1$
Conductor $283$
Sign $-0.0414 + 0.999i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.231 − 0.972i)2-s + (0.100 − 0.994i)3-s + (−0.892 − 0.451i)4-s + (−0.979 + 0.199i)5-s + (−0.944 − 0.328i)6-s + (0.920 − 0.390i)7-s + (−0.645 + 0.763i)8-s + (−0.979 − 0.199i)9-s + (−0.0334 + 0.999i)10-s + (−0.892 + 0.451i)11-s + (−0.538 + 0.842i)12-s + (−0.944 − 0.328i)13-s + (−0.166 − 0.986i)14-s + (0.100 + 0.994i)15-s + (0.593 + 0.805i)16-s + (−0.166 + 0.986i)17-s + ⋯
L(s,χ)  = 1  + (0.231 − 0.972i)2-s + (0.100 − 0.994i)3-s + (−0.892 − 0.451i)4-s + (−0.979 + 0.199i)5-s + (−0.944 − 0.328i)6-s + (0.920 − 0.390i)7-s + (−0.645 + 0.763i)8-s + (−0.979 − 0.199i)9-s + (−0.0334 + 0.999i)10-s + (−0.892 + 0.451i)11-s + (−0.538 + 0.842i)12-s + (−0.944 − 0.328i)13-s + (−0.166 − 0.986i)14-s + (0.100 + 0.994i)15-s + (0.593 + 0.805i)16-s + (−0.166 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0414 + 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0414 + 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.0414 + 0.999i$
Motivic weight: \(0\)
Character: $\chi_{283} (134, \cdot )$
Sato-Tate group: $\mu(47)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.0414 + 0.999i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.2139581653 - 0.2230134876i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.2139581653 - 0.2230134876i\)
\(L(\chi,1)\) \(\approx\) \(0.4273797608 - 0.5410195849i\)
\(L(1,\chi)\) \(\approx\) \(0.4273797608 - 0.5410195849i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.46145215624390213349822602959, −25.32794267678162066334305534133, −24.33459385336899426128240346723, −23.644145699443529677891365836067, −22.769570984634674565837046499304, −21.736338781035604002975125237050, −21.13283301422002849545875564209, −20.02545256081307847075602421436, −18.825128056606166824791483125387, −17.820089055943115229753874882436, −16.68180094880958151871057545857, −16.08152992219903236155602629764, −15.15950390536399242270646357896, −14.67865854711544717795207394272, −13.586664030848921328240046064704, −12.168184132823028329536817932361, −11.36403351026276939385072128789, −10.06947821731263450805635059382, −8.82874968565054260761616431887, −8.19051865710615723120209439718, −7.285770709988951505752558917435, −5.61054334804764926737445837017, −4.84410991783048062234270408099, −4.096190771262095233062118779021, −2.77556429423094345861520231941, 0.182081081347283000633629481117, 1.81341362755435770438142482260, 2.7453202030403290698864897599, 4.151185326275978375874442404370, 5.08312653160752701096006733355, 6.63156240571392773366427051370, 8.09257662376999862109132287924, 8.19717057510305074253299768944, 10.09971209377916725726894101391, 10.99415460805786345060899943455, 11.87319148122742879793200988151, 12.607063985343561194435651956913, 13.464935713826922349606488376991, 14.66025192828939363809316628533, 15.14186761413100786703978027715, 17.109053136395036950357456163708, 17.830341672381265369038189094694, 18.74316729295499821565138817757, 19.52795388813510568965179660933, 20.18340810260210289884242220893, 21.06306895189615174705902224871, 22.31197555936965342547070926028, 23.2282622696291588201145549042, 23.84371130147359394324292327273, 24.42560606013251013766224154246

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