Properties

Degree 1
Conductor 151
Sign $-0.999 + 0.00227i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.669 + 0.743i)2-s + (−0.929 + 0.368i)3-s + (−0.104 + 0.994i)4-s + (0.228 + 0.973i)5-s + (−0.895 − 0.444i)6-s + (−0.855 + 0.518i)7-s + (−0.809 + 0.587i)8-s + (0.728 − 0.684i)9-s + (−0.570 + 0.821i)10-s + (0.783 − 0.621i)11-s + (−0.268 − 0.963i)12-s + (−0.999 − 0.0418i)13-s + (−0.957 − 0.289i)14-s + (−0.570 − 0.821i)15-s + (−0.978 − 0.207i)16-s + (−0.0209 + 0.999i)17-s + ⋯
L(s,χ)  = 1  + (0.669 + 0.743i)2-s + (−0.929 + 0.368i)3-s + (−0.104 + 0.994i)4-s + (0.228 + 0.973i)5-s + (−0.895 − 0.444i)6-s + (−0.855 + 0.518i)7-s + (−0.809 + 0.587i)8-s + (0.728 − 0.684i)9-s + (−0.570 + 0.821i)10-s + (0.783 − 0.621i)11-s + (−0.268 − 0.963i)12-s + (−0.999 − 0.0418i)13-s + (−0.957 − 0.289i)14-s + (−0.570 − 0.821i)15-s + (−0.978 − 0.207i)16-s + (−0.0209 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.00227i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.00227i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $-0.999 + 0.00227i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (74, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ -0.999 + 0.00227i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.001027284643 + 0.9017513577i$
$L(\frac12,\chi)$  $\approx$  $0.001027284643 + 0.9017513577i$
$L(\chi,1)$  $\approx$  0.6041500947 + 0.7480270953i
$L(1,\chi)$  $\approx$  0.6041500947 + 0.7480270953i

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

−27.919139473338526304309722939131, −27.088558449226191787156735818123, −25.083888023283230192737085522, −24.52806170406313387862332683641, −23.28333076127415177061890133197, −22.78265495379109802308739129642, −21.8011488114409957143192056141, −20.700265143407479950380049050396, −19.701329683262127319149102563322, −18.92136364550849437289782725875, −17.40041270043748691595695558203, −16.741853905177320667230657787029, −15.502905319170730458026375355524, −13.99040871740033501005968687236, −12.9478178810820943576247951340, −12.37936807141526779022351905737, −11.453091913062916496128908491, −10.04721459824612462389078308237, −9.388297374454951973416120149185, −7.22549627380901796414436489378, −6.13298323516225258939633714433, −4.97966200025846536088354275752, −4.08536986055277827758932138307, −2.1308793917939336055184404990, −0.69538322375362203339309404605, 2.80562597209230735805687733189, 3.9904763813524169091780303519, 5.402619681706238882680043772724, 6.43641537358000254939635791953, 6.90416893958669801320087765115, 8.798080510665979343916978022321, 10.07538376500905149285141002829, 11.30956020711332142445937477697, 12.31260615110950119264363725471, 13.31670090371838879615647130576, 14.84685422921784532616422331360, 15.22700376615982853177361026954, 16.64463430250116178511890735626, 17.155948494534493483948086052184, 18.37199986885825907449100321672, 19.4644356089507756034545883532, 21.45307924585944266670797348706, 21.92620279283521136268829728308, 22.54184390754454985959285391053, 23.468919140270681885575298018, 24.54353823634414261393747066383, 25.59098404043940562868539352799, 26.54186633786817438332219914785, 27.28925083038127853114458435872, 28.693306982521830460532027258067

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