Properties

Degree 1
Conductor 151
Sign $0.371 + 0.928i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.669 − 0.743i)2-s + (−0.637 + 0.770i)3-s + (−0.104 − 0.994i)4-s + (−0.756 + 0.653i)5-s + (0.146 + 0.989i)6-s + (0.387 + 0.921i)7-s + (−0.809 − 0.587i)8-s + (−0.187 − 0.982i)9-s + (−0.0209 + 0.999i)10-s + (−0.348 + 0.937i)11-s + (0.832 + 0.553i)12-s + (−0.268 + 0.963i)13-s + (0.944 + 0.328i)14-s + (−0.0209 − 0.999i)15-s + (−0.978 + 0.207i)16-s + (0.604 + 0.796i)17-s + ⋯
L(s,χ)  = 1  + (0.669 − 0.743i)2-s + (−0.637 + 0.770i)3-s + (−0.104 − 0.994i)4-s + (−0.756 + 0.653i)5-s + (0.146 + 0.989i)6-s + (0.387 + 0.921i)7-s + (−0.809 − 0.587i)8-s + (−0.187 − 0.982i)9-s + (−0.0209 + 0.999i)10-s + (−0.348 + 0.937i)11-s + (0.832 + 0.553i)12-s + (−0.268 + 0.963i)13-s + (0.944 + 0.328i)14-s + (−0.0209 − 0.999i)15-s + (−0.978 + 0.207i)16-s + (0.604 + 0.796i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.371 + 0.928i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.371 + 0.928i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $0.371 + 0.928i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (58, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ 0.371 + 0.928i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7507806512 + 0.5083699242i$
$L(\frac12,\chi)$  $\approx$  $0.7507806512 + 0.5083699242i$
$L(\chi,1)$  $\approx$  0.9600322000 + 0.1404428618i
$L(1,\chi)$  $\approx$  0.9600322000 + 0.1404428618i

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

−27.56826900395791454736572294495, −27.06838059501056428505095681403, −25.59875242324014602499078423081, −24.54021351433592493398442240463, −23.90790049545896756188539736048, −23.27308384114053809654311456517, −22.42821912930726856396949721412, −21.065843972027020338901757240094, −20.056947576047448677161980083003, −18.78881277263378270747368484263, −17.528374348810225356049858001951, −16.7607462087817635804864748323, −16.02946208042190998338957215798, −14.693382171206581110968580477110, −13.4336954755932458715432582926, −12.8454135817607812365053889329, −11.718230885258338162064771144264, −10.7978019578664967559853358242, −8.52641884805441075949816769937, −7.73928959960788482329922675857, −6.87619326286419291961570642425, −5.425341685075634687673586886428, −4.65370926051976312162981428862, −3.103606060667939463014118324847, −0.68671485541656639358922981159, 2.097100615665542960572673276619, 3.58268502032679049574167658101, 4.56203198435842221185363580835, 5.63277340329661375709152147087, 6.89692362115835227118037582992, 8.80060460138397968977485544934, 10.08591752696239715241809266009, 10.90124967449462443665422784318, 11.953332108239367088154758939307, 12.42280875085358547917958542126, 14.40122972953511359202893117338, 15.06106367569773777607446853284, 15.75961585700692901190804430501, 17.33760259430366291023063285286, 18.5841146159627977002824559729, 19.30381006101269404451328704766, 20.71702302601232522555838111882, 21.47185034795626527608261246600, 22.20683650393888670534279497472, 23.32120276005057964063551676620, 23.63693010649132316784217862208, 25.292764980223087040986792780186, 26.669449704464705130398331475025, 27.589301746363431361214047606841, 28.24695244436999054572688055836

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