Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $0.564 - 0.825i$
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 + 3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s − 8-s + 9-s + 10-s − 11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s − 19-s + (0.5 − 0.866i)20-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s − 8-s + 9-s + 10-s − 11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s − 19-s + (0.5 − 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.564 - 0.825i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.564 - 0.825i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $0.564 - 0.825i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (30, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (0:\ ),\ 0.564 - 0.825i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.416746436 - 0.7475049420i$
$L(\frac12,\chi)$  $\approx$  $1.416746436 - 0.7475049420i$
$L(\chi,1)$  $\approx$  1.480275767 - 0.5913559350i
$L(1,\chi)$  $\approx$  1.480275767 - 0.5913559350i

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

−30.936542495419035073436867018409, −29.86020174475135715459882430440, −28.42220761368312932703120716788, −27.06440176744473298464818739646, −25.98499274823553872782633064153, −25.387415818410777981181485884092, −24.26210918700640844318644260812, −23.675937155318457915773871154711, −21.92386315912675631688682124813, −21.11231320480945187073893833150, −20.20837368248074191005930142830, −18.65947358549327828111895011823, −17.455958815341314448815498098, −16.261976141439401123708515018, −15.31349735134922349780519054073, −14.2121155988687000464206222705, −13.13123589870587120196439665342, −12.580892626736500488431426622894, −10.20879454770999193258961332186, −8.74304425017871328967106674875, −8.19676391262082818970669309976, −6.661244601694606540860893023915, −5.16533463723868436390471914110, −3.995950546828255445929154492411, −2.30342286488688634131709542324, 2.11791225206003671105047077571, 2.92937457628219937692608693705, 4.366494305618967868819839591695, 6.04931373259129114264067046151, 7.66894085391844235419971409167, 9.29918526046122057684128127314, 10.18283591305966238016318447076, 11.30092963531438220724969726030, 13.007301080427015204396665057546, 13.66480088781946990935857199040, 14.72610173722950657120586597020, 15.60123879594537243136559136706, 17.880964244777962659021410439622, 18.70979333040350496299242647859, 19.63185808584389387147749217864, 20.849699751403509936044224136890, 21.47424480773871067924746040152, 22.6233221815822852264963149120, 23.785870344676766797520027615239, 25.04643447669102090651299086547, 26.19774230085903690626672159077, 26.99955105701943542331515685658, 28.36207580792222761335636843424, 29.6760581412256614446742891258, 30.111567003434645332055045872493

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