Properties

Degree 1
Conductor $ 7^{2} $
Sign $-0.180 + 0.983i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0747 + 0.997i)2-s + (0.955 + 0.294i)3-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (0.826 + 0.563i)9-s + (−0.733 − 0.680i)10-s + (0.826 − 0.563i)11-s + (−0.988 − 0.149i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s,χ)  = 1  + (0.0747 + 0.997i)2-s + (0.955 + 0.294i)3-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (0.826 + 0.563i)9-s + (−0.733 − 0.680i)10-s + (0.826 − 0.563i)11-s + (−0.988 − 0.149i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.180 + 0.983i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.180 + 0.983i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $-0.180 + 0.983i$
motivic weight  =  \(0\)
character  :  $\chi_{49} (9, \cdot )$
Sato-Tate  :  $\mu(21)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 49,\ (0:\ ),\ -0.180 + 0.983i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6215422964 + 0.7461063056i$
$L(\frac12,\chi)$  $\approx$  $0.6215422964 + 0.7461063056i$
$L(\chi,1)$  $\approx$  0.8941903223 + 0.6667823158i
$L(1,\chi)$  $\approx$  0.8941903223 + 0.6667823158i

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

−33.050419127018849291811551559948, −31.66983281947063958025538686705, −31.46567552674061170748760079027, −30.131922132806117281779595984275, −29.15764850041286239214069589606, −27.59009391677608301906063798394, −27.03193801327451585101686334753, −25.3952546797193109641880564360, −24.13744133472030256425483977421, −22.953698928346765186115568942366, −21.41395226716727632966091140546, −20.285469144156621709858776157711, −19.61828974924320153022841107854, −18.60393323226349988787207209608, −16.953517507083550132858900930792, −15.0700471256074107720615258495, −13.9677101210085745767616755592, −12.56590217723395437377606096137, −11.809509776643290623425627911016, −9.77507150179969122072786599473, −8.823908334926926098982783810664, −7.43168652526564684449745036150, −4.699081667584628521138836757287, −3.456239851768577080933030946163, −1.66120189701890085511173596314, 3.20609248834328382833663615757, 4.50129761572855002434912630862, 6.6222983166209683958116040400, 7.84508320030159829778493793470, 8.92000199084025167245464411045, 10.46373422870415167549428655519, 12.560435470748406590762797966470, 14.15558590051888667775178865296, 14.86587965245352911219539018652, 15.85172560609709720951199265470, 17.26812926857122366541325120641, 18.94160418323464753567156365082, 19.6301004022053841555019424292, 21.57533839287893341219052462235, 22.503394330100626386590599281109, 23.94197961321592667493942042088, 24.94594218920471627265547408482, 26.12501733536833379423283729062, 26.921648412207451261975657978997, 27.73927594715139781661274750414, 30.135891608855586420349522805, 30.91330230915581376303564273965, 32.14269512298520859352989863229, 32.727204822126194225193426305633, 34.302293760013111500644272882216

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