Properties

Degree 1
Conductor 43
Sign $-0.263 - 0.964i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.623 − 0.781i)2-s + (0.988 + 0.149i)3-s + (−0.222 + 0.974i)4-s + (−0.826 − 0.563i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + (0.900 − 0.433i)8-s + (0.955 + 0.294i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (−0.365 + 0.930i)12-s + (0.0747 − 0.997i)13-s + (−0.988 + 0.149i)14-s + (−0.733 − 0.680i)15-s + (−0.900 − 0.433i)16-s + (0.826 − 0.563i)17-s + ⋯
L(s,χ)  = 1  + (−0.623 − 0.781i)2-s + (0.988 + 0.149i)3-s + (−0.222 + 0.974i)4-s + (−0.826 − 0.563i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + (0.900 − 0.433i)8-s + (0.955 + 0.294i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (−0.365 + 0.930i)12-s + (0.0747 − 0.997i)13-s + (−0.988 + 0.149i)14-s + (−0.733 − 0.680i)15-s + (−0.900 − 0.433i)16-s + (0.826 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.263 - 0.964i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.263 - 0.964i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.263 - 0.964i$
motivic weight  =  \(0\)
character  :  $\chi_{43} (3, \cdot )$
Sato-Tate  :  $\mu(42)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 43,\ (1:\ ),\ -0.263 - 0.964i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7835315006 - 1.026077192i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7835315006 - 1.026077192i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8590070432 - 0.5284962121i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8590070432 - 0.5284962121i\)

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

−34.66435933762802967469335337291, −33.82635113591973475295438317032, −32.291870485819122217976605586915, −31.28876918656780617572117586230, −30.39261256536673967847883785952, −28.3678538399499755265746082467, −27.38892335831282036093447831678, −26.22650140990890989135213811035, −25.47399327368679994798430078051, −24.2078298052200845853833632541, −23.23032112364413897763727021631, −21.3726861314354851926534288585, −19.75010082695519795075796233103, −18.8898398257357005732102802018, −17.94541360226905347920374140875, −16.00990310695754597112532776498, −14.96866828151609204320158883410, −14.3243200097483715999947916970, −12.24706247662790428888152262115, −10.373530276766533722633293362866, −8.84881819835357165462126843750, −7.88851318727036878849788131728, −6.65916750559902502853374407725, −4.414171762397220918161585504407, −2.11807725919687289196571399375, 0.9617534275120312872638837617, 3.14954683593380006041750365016, 4.36156254087981038774846773489, 7.81420732861112439603099171278, 8.241043117129554897157847483, 9.89334085401403754372255488592, 11.18114996438025052094451232322, 12.74828744374762248377599455066, 13.93679974961547816907913185512, 15.727864212645197280362939638785, 16.93917896012103006272987470601, 18.62134097204783614291976810027, 19.70401133865676755788849456570, 20.48382094228772132963520089143, 21.41077685844965669057134210605, 23.28863369991328083689025001849, 24.72954354871513648401912087711, 26.07240098883164244028990025337, 27.25837829522895906076585036227, 27.61958010230680284283220748593, 29.57538925546914796058843275105, 30.360643087729561105335696935535, 31.59901610109045480282424928936, 32.343725882738698862031064635626, 34.28398670599992202636531944933

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