Properties

Degree 1
Conductor $ 7^{2} $
Sign $0.740 + 0.672i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.623 + 0.781i)2-s + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (0.623 + 0.781i)11-s + (0.222 + 0.974i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 18-s − 19-s + ⋯
L(s,χ)  = 1  + (0.623 + 0.781i)2-s + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (0.623 + 0.781i)11-s + (0.222 + 0.974i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 18-s − 19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.740 + 0.672i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.740 + 0.672i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $0.740 + 0.672i$
motivic weight  =  \(0\)
character  :  $\chi_{49} (20, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 49,\ (1:\ ),\ 0.740 + 0.672i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.711655742 + 1.047561695i$
$L(\frac12,\chi)$  $\approx$  $2.711655742 + 1.047561695i$
$L(\chi,1)$  $\approx$  1.933522022 + 0.5738591870i
$L(1,\chi)$  $\approx$  1.933522022 + 0.5738591870i

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

−33.01560139295338333624237958665, −32.1585639030322079227528821369, −31.17853261586234143853442850195, −29.96572782187551974006290702295, −29.235582475837828427823084306708, −27.6491798946677525058519066472, −26.5921531224977351772189947951, −25.23095736517513893733739845866, −24.16045647030041821108524918740, −22.32777274430256496607072567858, −21.63298590356526447955803139245, −20.67080485422128521366183663117, −19.40876700595867945637524705065, −18.45506147970721820399419395353, −16.480519454042238572028363106059, −14.62161504329244519139432522169, −14.18847625044676389670844948985, −12.94147629736310746170412329113, −11.17148374710816495178862994010, −9.91626169415771365371355080226, −8.95429599466715683624038432701, −6.5796101135264379510338645468, −4.8227586467431939975791017115, −3.24437793919478916539078001260, −1.96323991504187471942015450856, 2.13236234001519963484114991641, 3.999754848446052884324849861623, 5.74996501742913203102593225399, 7.16641085502735283874832682399, 8.49668518900253679599271469358, 9.73285854943002944160326683542, 12.41914299114577662934123956293, 13.139893037680417360428596609547, 14.40443193621918326798585861609, 15.23411472443455792064336297697, 17.04334488761349903628423445064, 17.85393097478600669665774001469, 19.68545494722767291326778875470, 20.92051115222391578605172456948, 21.93014023991814398008288745123, 23.46352958972426754175324641674, 24.67043890390799255253186107425, 25.330422012179091899533939523108, 26.1706684691188098121106605213, 27.72983058329608430044130366313, 29.65931081922095993237119792697, 30.31869346408141019386676508977, 31.681459882926586810158890675026, 32.46733635813848866039795340910, 33.352536735586069059405344435

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