Properties

Degree 1
Conductor 151
Sign $-0.485 + 0.874i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.104 − 0.994i)2-s + (−0.187 − 0.982i)3-s + (−0.978 + 0.207i)4-s + (0.146 − 0.989i)5-s + (−0.957 + 0.289i)6-s + (−0.699 + 0.714i)7-s + (0.309 + 0.951i)8-s + (−0.929 + 0.368i)9-s + (−0.999 − 0.0418i)10-s + (−0.756 − 0.653i)11-s + (0.387 + 0.921i)12-s + (−0.855 − 0.518i)13-s + (0.783 + 0.621i)14-s + (−0.999 + 0.0418i)15-s + (0.913 − 0.406i)16-s + (−0.268 + 0.963i)17-s + ⋯
L(s,χ)  = 1  + (−0.104 − 0.994i)2-s + (−0.187 − 0.982i)3-s + (−0.978 + 0.207i)4-s + (0.146 − 0.989i)5-s + (−0.957 + 0.289i)6-s + (−0.699 + 0.714i)7-s + (0.309 + 0.951i)8-s + (−0.929 + 0.368i)9-s + (−0.999 − 0.0418i)10-s + (−0.756 − 0.653i)11-s + (0.387 + 0.921i)12-s + (−0.855 − 0.518i)13-s + (0.783 + 0.621i)14-s + (−0.999 + 0.0418i)15-s + (0.913 − 0.406i)16-s + (−0.268 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.485 + 0.874i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.485 + 0.874i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(151\)
\( \varepsilon \)  =  $-0.485 + 0.874i$
motivic weight  =  \(0\)
character  :  $\chi_{151} (42, \cdot )$
Sato-Tate  :  $\mu(75)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 151,\ (0:\ ),\ -0.485 + 0.874i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2184310190 - 0.3709152306i$
$L(\frac12,\chi)$  $\approx$  $-0.2184310190 - 0.3709152306i$
$L(\chi,1)$  $\approx$  0.3278402740 - 0.5301644378i
$L(1,\chi)$  $\approx$  0.3278402740 - 0.5301644378i

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

−28.633457070896520697561423794086, −27.15922245914093769760581240143, −26.72140994363353897568855803289, −25.994489346221707858586179620394, −25.13139184797220027984626513330, −23.564224589791583019704107851262, −22.7479535731531510409537766270, −22.31237483464008998023810637305, −21.03822980395725649685989763423, −19.71949679922893908002109847699, −18.49474641364724895036774809961, −17.54352171488047076257912601999, −16.48450791603018264698121441881, −15.80548367910761290278864257214, −14.631878579867717835098730540901, −14.10362725907505509002358697803, −12.61132304372409829767932645159, −10.823582128708787631133475094429, −10.006074758413966389339200895420, −9.254756179135579870792052957766, −7.4954161438952126998004320772, −6.73551247753168683767915455661, −5.40317116220394872590151260917, −4.28288371293591921554177047857, −2.99956052014800567445778061308, 0.376282232717851485749560592920, 1.99029790358285386786175731159, 3.09016993163124896679221146001, 5.0130721972837192346213921701, 5.90110258206780848458880093, 7.78522445720674686876674475640, 8.72042404722939872479186940450, 9.739726921323168624911631490894, 11.22027248777731434578594133698, 12.18786365779026608546610210180, 13.09330281841582755782864333953, 13.403361972618336595507638426110, 15.249299270804079389272497999933, 16.81964569322780004468402580523, 17.55892296122111429064800223893, 18.703922812836758838531816742656, 19.44249254030219857821207366002, 20.23538739746700679701658196402, 21.518975601882665730061647713151, 22.29091226911261098311902943026, 23.53854117003412631965970010079, 24.32825979545533496422525225206, 25.405103112532647122678675287850, 26.49413809067053510059958846119, 27.949542470925343589670904924614

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