Properties

Degree 1
Conductor 151
Sign $-0.531 - 0.847i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.978 + 0.207i)2-s + (0.535 − 0.844i)3-s + (0.913 − 0.406i)4-s + (−0.570 + 0.821i)5-s + (−0.348 + 0.937i)6-s + (−0.957 − 0.289i)7-s + (−0.809 + 0.587i)8-s + (−0.425 − 0.904i)9-s + (0.387 − 0.921i)10-s + (0.463 − 0.886i)11-s + (0.146 − 0.989i)12-s + (−0.895 − 0.444i)13-s + (0.996 + 0.0836i)14-s + (0.387 + 0.921i)15-s + (0.669 − 0.743i)16-s + (0.228 − 0.973i)17-s + ⋯
L(s,χ)  = 1  + (−0.978 + 0.207i)2-s + (0.535 − 0.844i)3-s + (0.913 − 0.406i)4-s + (−0.570 + 0.821i)5-s + (−0.348 + 0.937i)6-s + (−0.957 − 0.289i)7-s + (−0.809 + 0.587i)8-s + (−0.425 − 0.904i)9-s + (0.387 − 0.921i)10-s + (0.463 − 0.886i)11-s + (0.146 − 0.989i)12-s + (−0.895 − 0.444i)13-s + (0.996 + 0.0836i)14-s + (0.387 + 0.921i)15-s + (0.669 − 0.743i)16-s + (0.228 − 0.973i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.531 - 0.847i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.531 - 0.847i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $-0.531 - 0.847i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (69, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ -0.531 - 0.847i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2284366628 - 0.4128143497i$
$L(\frac12,\chi)$  $\approx$  $0.2284366628 - 0.4128143497i$
$L(\chi,1)$  $\approx$  0.5516954585 - 0.2100565294i
$L(1,\chi)$  $\approx$  0.5516954585 - 0.2100565294i

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

−28.3325242039299450740330458870, −27.36018271934824149695181494835, −26.66258016352028987784866023235, −25.48584882773113627820749384819, −25.07117223911418263152421205427, −23.614309100028923518743670996329, −22.15942617452661690914366988777, −21.284261235373957220720619943680, −20.07853795992634904497073055612, −19.73424574549130172761152469529, −18.77470267095991267820355877948, −16.9664901129481276794961588970, −16.61370953095911609417775813240, −15.50012670769659301883450361277, −14.71549299596366478906877261045, −12.71210673620290386040123693744, −12.077512008361738121723104670648, −10.5119684193107861277896340578, −9.6471803139553103063606938567, −8.846176814430675838933104449411, −7.90201547620774717514968763736, −6.47191338103741911821039526085, −4.61146555983553571866575800541, −3.452952641985731224882400680909, −1.99997264314392252183873103213, 0.49053416107113822339889283408, 2.50419011030176039748477844273, 3.38773523017351541335420856089, 6.04814377458962052072331510975, 6.99544614068716662963744646831, 7.65914524678744828316149678597, 8.91163708358687360079760260993, 9.95115462442790039061873490157, 11.24507672796740660691484427352, 12.20305857087974441715644264089, 13.65016621005835321377138496015, 14.72750948465630692412623269034, 15.680403757994383976122475614494, 16.86486480448154046710659389457, 17.96287465798511733509474076116, 18.9683248948252236029406150228, 19.486076922813356394473649135150, 20.14602781256930849474274336180, 21.837286304842237337726910017222, 23.15763126479518368725831268979, 23.97408990543632611364719177323, 25.13161185502167195524971154715, 25.7499589423164137407986335543, 26.773742769754367095753436634, 27.32243564817032324403957868256

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