Properties

Degree 1
Conductor $ 7^{2} $
Sign $0.991 + 0.127i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.733 − 0.680i)2-s + (−0.988 − 0.149i)3-s + (0.0747 + 0.997i)4-s + (0.365 + 0.930i)5-s + (0.623 + 0.781i)6-s + (0.623 − 0.781i)8-s + (0.955 + 0.294i)9-s + (0.365 − 0.930i)10-s + (0.955 − 0.294i)11-s + (0.0747 − 0.997i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.988 + 0.149i)16-s + (0.826 + 0.563i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s,χ)  = 1  + (−0.733 − 0.680i)2-s + (−0.988 − 0.149i)3-s + (0.0747 + 0.997i)4-s + (0.365 + 0.930i)5-s + (0.623 + 0.781i)6-s + (0.623 − 0.781i)8-s + (0.955 + 0.294i)9-s + (0.365 − 0.930i)10-s + (0.955 − 0.294i)11-s + (0.0747 − 0.997i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.988 + 0.149i)16-s + (0.826 + 0.563i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.991 + 0.127i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.991 + 0.127i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $0.991 + 0.127i$
motivic weight  =  \(0\)
character  :  $\chi_{49} (46, \cdot )$
Sato-Tate  :  $\mu(21)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 49,\ (0:\ ),\ 0.991 + 0.127i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4975307434 + 0.03194253354i$
$L(\frac12,\chi)$  $\approx$  $0.4975307434 + 0.03194253354i$
$L(\chi,1)$  $\approx$  0.6102811862 - 0.03559613097i
$L(1,\chi)$  $\approx$  0.6102811862 - 0.03559613097i

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

−33.90858231502444002250131694913, −32.79806881977478916150950057615, −32.21607122074541505699792745829, −29.89849911154759148429711258690, −28.90665059899623230325274179987, −27.77906770635638706556955703985, −27.35548018932918913598163409390, −25.50939247539995804151512450769, −24.61862782970093631143752966176, −23.53910142859883291808264941063, −22.37984779019754235614273229877, −20.78479013667969903631823792390, −19.43078038047820029136980327412, −17.83201905961061607923958775092, −17.14493338591606630769369784805, −16.21572150580853718283713191383, −14.93103467721236441065818513028, −13.07471934580024541880643963320, −11.59026146725713594453683091431, −10.08242793757184730327576037019, −9.04230079135110757187065401777, −7.29362547278191517820874441276, −5.84240042053834901489450448073, −4.79919286644587074478368981611, −1.1471600810512843307324139808, 1.78115653017227131410547689758, 3.86852105993676888812500876133, 6.18360925461122332147156635021, 7.36937270918464454681646411709, 9.370964178769244681604074071930, 10.613055392998053723355596121914, 11.51013850042803902603025228287, 12.72123722564523985542627491722, 14.47206184594732202531623351211, 16.532188310884843729015608980282, 17.24063820327101433299942508638, 18.59328269841454580966174552215, 19.17363064396215053751386493372, 21.15783088627955628892528882617, 21.99871215242018260803494001041, 23.009283445749418963281014157, 24.71829837801189482098432821416, 26.074369271781389671758345400655, 27.17863782032006917269889699765, 28.13168378121628631351523862342, 29.452011562929859176356143811717, 29.852326389539635081215163265833, 31.1066065814952067253425453952, 33.07393363087017020425091235792, 34.18539390577231514965402256673

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