Properties

Degree 1
Conductor 149
Sign $0.580 + 0.814i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.996 + 0.0848i)2-s + (−0.450 + 0.892i)3-s + (0.985 − 0.169i)4-s + (−0.828 − 0.559i)5-s + (0.372 − 0.927i)6-s + (0.873 + 0.487i)7-s + (−0.967 + 0.251i)8-s + (−0.594 − 0.803i)9-s + (0.873 + 0.487i)10-s + (0.985 + 0.169i)11-s + (−0.292 + 0.956i)12-s + (0.210 − 0.977i)13-s + (−0.911 − 0.411i)14-s + (0.873 − 0.487i)15-s + (0.942 − 0.333i)16-s + (−0.450 + 0.892i)17-s + ⋯
L(s,χ)  = 1  + (−0.996 + 0.0848i)2-s + (−0.450 + 0.892i)3-s + (0.985 − 0.169i)4-s + (−0.828 − 0.559i)5-s + (0.372 − 0.927i)6-s + (0.873 + 0.487i)7-s + (−0.967 + 0.251i)8-s + (−0.594 − 0.803i)9-s + (0.873 + 0.487i)10-s + (0.985 + 0.169i)11-s + (−0.292 + 0.956i)12-s + (0.210 − 0.977i)13-s + (−0.911 − 0.411i)14-s + (0.873 − 0.487i)15-s + (0.942 − 0.333i)16-s + (−0.450 + 0.892i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.580 + 0.814i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.580 + 0.814i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(149\)
\( \varepsilon \)  =  $0.580 + 0.814i$
motivic weight  =  \(0\)
character  :  $\chi_{149} (37, \cdot )$
Sato-Tate  :  $\mu(37)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 149,\ (0:\ ),\ 0.580 + 0.814i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5304062275 + 0.2734416131i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5304062275 + 0.2734416131i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5962167385 + 0.1719754171i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5962167385 + 0.1719754171i\)

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

−27.8844664375054023853872882563, −27.0395369078415714428488942902, −26.31880783345384802704315141856, −24.821671643451448294781777493004, −24.32549405785386262071109865641, −23.255557027237895883336411925248, −22.19867508408045279750500040293, −20.63389405702668436036843868794, −19.77073349180708275980344412988, −18.734279750766274174038423979169, −18.26282648830568449431225091298, −17.017957456692500188084845480697, −16.37100698458344015972355446814, −14.84118158918162647574188243540, −13.842273302014795020414705081858, −11.9806379047846670849877635024, −11.54791552114373899415742751574, −10.72060311585134995445758191241, −9.05758325267417024676012051596, −7.83724580095365960362965201553, −7.180722036129912290605719242718, −6.177676484571305483933013483419, −4.140074915512178834495376439526, −2.34332951174947892119950891371, −0.95422900880389767147815041003, 1.22348371217601972082482078442, 3.34342535433604821912799513746, 4.78778216298590091290443854321, 5.95953751937440664192934748418, 7.55926369033649807602821565752, 8.67567027646132236021316080081, 9.37446526643948066735593576045, 10.88794852112787170581319837444, 11.45099045385562715838434801193, 12.44851229028835989959691636402, 14.75570941400123012204106473428, 15.37923002705433019008747846962, 16.24091317951677079330747755920, 17.35992835010791435800296286844, 17.90483112068823247119419954590, 19.53463230348236324354004207222, 20.180439653169309214358506552501, 21.15496802707293159229753503482, 22.21642077127887771808002483388, 23.577581942681178587936361093217, 24.430073887773652810465117227016, 25.4589482249739963136117880738, 26.72588606480753436529895156288, 27.49591520125796773456917035032, 27.93023287810796421104540688571

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