Properties

Degree 1
Conductor 131
Sign $-0.584 - 0.811i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.262 − 0.964i)2-s + (0.958 − 0.285i)3-s + (−0.861 + 0.506i)4-s + (−0.943 − 0.331i)5-s + (−0.527 − 0.849i)6-s + (0.926 − 0.377i)7-s + (0.715 + 0.698i)8-s + (0.836 − 0.548i)9-s + (−0.0724 + 0.997i)10-s + (−0.681 − 0.732i)11-s + (−0.681 + 0.732i)12-s + (−0.0724 − 0.997i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (0.485 − 0.873i)16-s + (−0.906 + 0.421i)17-s + ⋯
L(s,χ)  = 1  + (−0.262 − 0.964i)2-s + (0.958 − 0.285i)3-s + (−0.861 + 0.506i)4-s + (−0.943 − 0.331i)5-s + (−0.527 − 0.849i)6-s + (0.926 − 0.377i)7-s + (0.715 + 0.698i)8-s + (0.836 − 0.548i)9-s + (−0.0724 + 0.997i)10-s + (−0.681 − 0.732i)11-s + (−0.681 + 0.732i)12-s + (−0.0724 − 0.997i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (0.485 − 0.873i)16-s + (−0.906 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.584 - 0.811i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.584 - 0.811i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $-0.584 - 0.811i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (25, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ -0.584 - 0.811i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4746884886 - 0.9273660449i$
$L(\frac12,\chi)$  $\approx$  $0.4746884886 - 0.9273660449i$
$L(\chi,1)$  $\approx$  0.7863425241 - 0.6765780780i
$L(1,\chi)$  $\approx$  0.7863425241 - 0.6765780780i

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

−28.64669372498539591869002510654, −27.57347618090339794211429671280, −26.82948226557908365472393936193, −26.222310021697856182996491647444, −25.049471761608317601005336327444, −24.24734651686956595342747998882, −23.331052546156015822492998214378, −22.14216518597764979043872548738, −20.88603016109956909028210348426, −19.79961878013041993070037900460, −18.68971684525597881122709570366, −18.07075183453534676917264995654, −16.39332030289510231510437278064, −15.57967186983493135833826806737, −14.72234411712236910240211305387, −14.10500148951097149438709742874, −12.59269670719336032648834177526, −11.01014009558118450379505197303, −9.71565978127944685111439568475, −8.50847268828578647974138067041, −7.84065527028669614107425329238, −6.82626258451210226782205705649, −4.861730146508515533333800191571, −4.093334390547271788010381213294, −2.15849980317827096311636747476, 1.036303426644573767826912359568, 2.653439871046725931769358402208, 3.78523114260525264586734166549, 4.92414442582714049248640922684, 7.53111884156312444692361116903, 8.16656063145271757704130360848, 9.048025891914000675272215615817, 10.6190668716814164008258235721, 11.458047802067830721695260843370, 12.81347008124064462590936579509, 13.49728465457928390984615961640, 14.785790795644787712465421701726, 15.89317544222510192942551772829, 17.551209627213141655756603528288, 18.36537182705084733409369253551, 19.62792806381835707033110844264, 20.019148218422784120139900296557, 20.96725692595274750977620250075, 21.955478939670475310322977681615, 23.5964835524188471786457271875, 24.09178361764799395153996670854, 25.57355473834443508482988771600, 26.84004328291787406569875227996, 27.1142401711802252690233384124, 28.26890392795039194088579983591

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