Properties

Degree 1
Conductor $ 13 \cdot 31 $
Sign $0.929 + 0.368i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 + 0.866i)18-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.929 + 0.368i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.929 + 0.368i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.929 + 0.368i$
motivic weight  =  \(0\)
character  :  $\chi_{403} (191, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 403,\ (0:\ ),\ 0.929 + 0.368i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8847224533 + 0.1691689094i$
$L(\frac12,\chi)$  $\approx$  $0.8847224533 + 0.1691689094i$
$L(\chi,1)$  $\approx$  0.7805416416 + 0.03625445636i
$L(1,\chi)$  $\approx$  0.7805416416 + 0.03625445636i

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

−24.39809702995557170554610192817, −23.72292367070727140581882852657, −23.00995965093261702534575116159, −21.99234987067715277554937386675, −20.56072975179844054732042948543, −19.68218099458472944798574110509, −18.957859354776103503447929547314, −17.916561333608548040820538464949, −17.34713592927311029658442532078, −16.56490189097513957267371558966, −15.80066881887721147379116131952, −14.449664950377928606782610760522, −13.9509792701791443474459570333, −12.67039963382176395582761975848, −11.78452943185868665187463455101, −11.01046569703415955618722644253, −9.52772655587826180480022663861, −8.59810051802033896016748621661, −7.705586393275866613971378658861, −7.20176945443742191002042342809, −5.73118610654433869640875082692, −5.229697298690938685994346129563, −4.00252950113873816352617352121, −1.641227731048053291020181168072, −0.96878510190201682482294987520, 1.07361895305273691432943458450, 2.67244188028637885874146802079, 3.78355942225480550328241031056, 4.43087565475953049541950434845, 5.78774011910077924837439495777, 7.23176228794710874457209185302, 8.182373836982065083046428176397, 9.28787321570502222038513649639, 10.18499764022580823742282785313, 10.95484189146921188827095981510, 11.73435605501912532399213892280, 12.147693568415667363025296260014, 14.04666424095745281950237812943, 14.55996686672905859794808239449, 15.720775674222909055517853804753, 16.75166003543751810909944294329, 17.50550508129891820536948667590, 18.31494950163523026348776675373, 19.15067984312762663293486114746, 20.27024704239796481658144589424, 20.84855972732194021936835492439, 21.845037234064837000646859037, 22.43400544749358879157828250928, 23.10476436449895950288015140032, 24.327177969359914300658975177544

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