Properties

Degree 1
Conductor $ 5^{2} \cdot 7 $
Sign $0.743 - 0.668i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.951 + 0.309i)13-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (−0.866 + 0.5i)18-s + (0.913 − 0.406i)19-s + (−0.587 − 0.809i)22-s + (−0.207 + 0.978i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s,χ)  = 1  + (0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.951 + 0.309i)13-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (−0.866 + 0.5i)18-s + (0.913 − 0.406i)19-s + (−0.587 − 0.809i)22-s + (−0.207 + 0.978i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.743 - 0.668i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.743 - 0.668i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(175\)    =    \(5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.743 - 0.668i$
motivic weight  =  \(0\)
character  :  $\chi_{175} (87, \cdot )$
Sato-Tate  :  $\mu(60)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 175,\ (0:\ ),\ 0.743 - 0.668i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9797356834 - 0.3754746417i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9797356834 - 0.3754746417i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9644375316 - 0.2738050804i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9644375316 - 0.2738050804i\)

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

−27.62888723248432350184982599245, −26.31226248080069941424487355309, −25.23467110904049414189483749197, −24.86428322398254778141164955264, −23.67316083671286632385706196042, −22.96051416868312839330105394139, −22.32267627014309838709104315791, −20.87880637136145921763750186236, −19.54181697260883308687219056757, −18.34606343788864219580842879292, −17.84029393272852577300893006295, −16.7326415128704686840507667258, −15.936459521831631197985218455478, −14.49305086103660622253177728428, −13.84966614422364655579423878851, −12.60980367436480750207658098755, −11.992306431937068742657561619716, −10.348161915998625828259974531120, −8.87884323516486944446832341981, −7.84820537560086676480917529368, −6.89051128254044758069299057910, −5.98074755432580654932757996615, −4.89924841055472443892533544472, −3.33208613500128635950758953479, −1.25579582554016724101579035613, 1.153922591516583840028484871834, 3.17509459750197425546593742920, 3.928526042596736706191642950937, 5.23399533252627905281514799607, 6.1878312021091750126474680406, 8.35964321649641854043577440104, 9.39899587898599298996893135550, 10.22225656838285018031429562739, 11.46070905129206708164386830559, 11.79494646582711032075991741034, 13.446248020562053105632907738358, 14.264762549884324171948309141656, 15.46608625588693029364927186819, 16.58050705137990438250454521295, 17.60942011524698062134216143039, 18.71730737556844110984952903520, 19.74873063981090145799540530390, 20.79938059248118932146659227030, 21.46204900896723960224692903372, 22.31058195949054914078662519519, 23.15836932776986208637314951544, 24.08213390232688284064118614590, 25.7281750325613110959411481324, 26.72799824608092510498775290082, 27.55989639009324196148622589682

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