Properties

Degree 1
Conductor $ 7 \cdot 571 $
Sign $0.457 - 0.889i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.821 − 0.569i)2-s + (−0.938 − 0.345i)3-s + (0.350 + 0.936i)4-s + (0.360 + 0.932i)5-s + (0.574 + 0.818i)6-s + (0.245 − 0.969i)8-s + (0.761 + 0.648i)9-s + (0.234 − 0.972i)10-s + (0.959 + 0.282i)11-s + (−0.00551 − 0.999i)12-s + (−0.0715 + 0.997i)13-s + (−0.0165 − 0.999i)15-s + (−0.754 + 0.656i)16-s + (−0.391 − 0.920i)17-s + (−0.256 − 0.966i)18-s + (0.556 + 0.831i)19-s + ⋯
L(s,χ)  = 1  + (−0.821 − 0.569i)2-s + (−0.938 − 0.345i)3-s + (0.350 + 0.936i)4-s + (0.360 + 0.932i)5-s + (0.574 + 0.818i)6-s + (0.245 − 0.969i)8-s + (0.761 + 0.648i)9-s + (0.234 − 0.972i)10-s + (0.959 + 0.282i)11-s + (−0.00551 − 0.999i)12-s + (−0.0715 + 0.997i)13-s + (−0.0165 − 0.999i)15-s + (−0.754 + 0.656i)16-s + (−0.391 − 0.920i)17-s + (−0.256 − 0.966i)18-s + (0.556 + 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.457 - 0.889i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.457 - 0.889i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(3997\)    =    \(7 \cdot 571\)
\( \varepsilon \)  =  $0.457 - 0.889i$
motivic weight  =  \(0\)
character  :  $\chi_{3997} (237, \cdot )$
Sato-Tate  :  $\mu(570)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 3997,\ (1:\ ),\ 0.457 - 0.889i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7715134394 - 0.4705033183i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7715134394 - 0.4705033183i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6135739523 - 0.05761313731i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6135739523 - 0.05761313731i\)

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

−17.935279775500569763576274962194, −17.66061295836011664839467953686, −17.1010236925884841447846421568, −16.59664331232190610585147756474, −15.866872588612262846511468758718, −15.32118460154293198117660911196, −14.68289112544601664016927803961, −13.53722693624063046212406333014, −13.03685971474725912164403454366, −12.00592579323681787949518071612, −11.48664912005200560444844005698, −10.73607949161671686605386378420, −9.96554277616946071021319810493, −9.462768005315687931588957120059, −8.75579772021984607825794801146, −8.10837648087182897785573008799, −7.06830579857831803117661453291, −6.50053131771622694168531625012, −5.51369649192720575308659578130, −5.427917237587704773102521261226, −4.42331687341416222773374344071, −3.52889761017018552425394603255, −2.07529696874488496846245244798, −1.1627272085825319756544762712, −0.68898155884270382642918496250, 0.30073939432251059717543231934, 1.3907148629811153055924953905, 1.92255345984161103134707843276, 2.79722657918345900407945513255, 3.79863411270022386120738398349, 4.52109816711243289544586396448, 5.65006333833684571867510754229, 6.502923866029961666564771866884, 7.0752959132060101706815119580, 7.362374599796194595702755785070, 8.59814231873089265799917732913, 9.44674113721715019417457116333, 9.861059088902146964934575505288, 10.85072403149959839748600216467, 11.12576989369409464436554909894, 11.89788258031857766928984330315, 12.37081491535459929708904038671, 13.22712337835356659720436036765, 14.12505918338212762144424963622, 14.63775413467393439128865727997, 15.9123933833656356599865996073, 16.35902495419399789845364829773, 17.01563689439541664743223404476, 17.704739844212593814124531663304, 18.17769499758166641025694873294

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