Properties

Degree 1
Conductor $ 3 \cdot 19 $
Sign $-0.174 + 0.984i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (−0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s,χ)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (−0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.174 + 0.984i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.174 + 0.984i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $-0.174 + 0.984i$
motivic weight  =  \(0\)
character  :  $\chi_{57} (41, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 57,\ (0:\ ),\ -0.174 + 0.984i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6077953449 + 0.7248225467i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6077953449 + 0.7248225467i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8667234833 + 0.6187931938i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8667234833 + 0.6187931938i\)

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

−32.57637132692121609218641907323, −31.49617754927088811083138206526, −30.01252662001282626912823223958, −29.37153805030093941494657499301, −28.52061249393585102429112572465, −27.082091569368029063124174743526, −26.13824452907088080542437243541, −24.51866768605796804806670313681, −23.35205339541211711847911886525, −21.96866382478602605973663556863, −21.30230994874703382983453916105, −19.94460725561293330310895586913, −19.10739717443278879797421764028, −17.520822273450884887224292282074, −16.67218854982305020240350808334, −14.42902316119806014162908844782, −13.54134656864630331000437040735, −12.50747860760458802056657309859, −10.93668149189960526621570116012, −9.852038647010820539245450152374, −8.75784792740989738424037575652, −6.483191541039964732611343358036, −4.88174169296693573692084506043, −3.31946694304791485919382994952, −1.475562779783102637766625746055, 2.72984526246235140767908668517, 4.89890724724105088675221851574, 6.114560457652120145987460866172, 7.26358900431775537839898248888, 9.075609591594104480090949829279, 9.914482947113240477285222565011, 12.20675917906107073595582094982, 13.34430131329149329136470919047, 14.63466326013451613180774809777, 15.52232901507429117980688754955, 17.04458715726933389795587854491, 17.89258795532597902830601364634, 19.0490108901227497937498717808, 20.94828960804328665618860159013, 22.347550134087659727931456431889, 22.68367261083105506071938291166, 24.686400262169207605574780363474, 25.12320566813421134975610222927, 26.1656481140313772885974404716, 27.429059141046289400719104958136, 28.69173770478102770660959590617, 30.01819112915587646756581950462, 31.2923123239067786000201799463, 32.32492135635258141239960093950, 33.285343262210117306952016833

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