Properties

Degree 1
Conductor $ 5 \cdot 7 \cdot 29 $
Sign $0.525 - 0.850i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  i·2-s i·3-s − 4-s − 6-s + i·8-s − 9-s − 11-s + i·12-s + i·13-s + 16-s + i·17-s + i·18-s − 19-s + i·22-s i·23-s + 24-s + ⋯
L(s,χ)  = 1  i·2-s i·3-s − 4-s − 6-s + i·8-s − 9-s − 11-s + i·12-s + i·13-s + 16-s + i·17-s + i·18-s − 19-s + i·22-s i·23-s + 24-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.525 - 0.850i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.525 - 0.850i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1015\)    =    \(5 \cdot 7 \cdot 29\)
\( \varepsilon \)  =  $0.525 - 0.850i$
motivic weight  =  \(0\)
character  :  $\chi_{1015} (202, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1015,\ (0:\ ),\ 0.525 - 0.850i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7893996699 - 0.4401191415i$
$L(\frac12,\chi)$  $\approx$  $0.7893996699 - 0.4401191415i$
$L(\chi,1)$  $\approx$  0.6617987563 - 0.4901941228i
$L(1,\chi)$  $\approx$  0.6617987563 - 0.4901941228i

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

−21.82312270229038325098672238246, −21.12155244239185611058401169742, −20.352130988457197948209956530019, −19.33653409188774657741983120653, −18.382902735584542668943075330012, −17.58226858278770227837069020792, −16.970254340832334515442765715131, −16.040796211726501266151174394926, −15.46291938717321707814669580790, −15.04214064192111604362649739676, −13.94786356283130053255968673301, −13.343637835344611674936379517714, −12.3335695053067711312366235807, −11.14314693318393228039738754884, −10.21818042481765528009723027441, −9.70709496509257508587564877482, −8.635623623379755183261526098468, −8.056083367345658165251909417193, −7.09699812708627397530798850557, −5.91346061175355082983153279693, −5.2748314604914725205454466223, −4.57683885132616220884987342120, −3.53741331405148931931332023380, −2.61346308486070065576045482854, −0.49901326454250295795399057453, 0.90236246237444444838549540776, 2.11036298731481838767081551925, 2.536597652512953362270693713194, 3.84145965634698139890705608392, 4.76801308236269482454618277637, 5.87803099805988477549518088457, 6.7060409974495331402589346032, 7.98134221751357215316203346037, 8.436973401305200677851067343964, 9.41801482876526472253984827707, 10.536571344383343234396699422116, 11.09958162152062798972129437729, 12.03670855608964686932570755367, 12.81090685711075339891211637847, 13.16864564186653137611331069653, 14.24360973387817197857553014081, 14.73799549408889094013565512028, 16.180393078370830736278844074729, 17.1518554542438585142878610356, 17.80778725257730295608294284089, 18.56461447465240711900469984670, 19.28643288870695925583589733765, 19.59739389386546247533189517628, 20.95355513208364414726079563964, 21.10432453846603928142181735359

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