Properties

Degree 1
Conductor 59
Sign $-0.996 + 0.0870i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.856 − 0.515i)2-s + (−0.370 − 0.928i)3-s + (0.468 − 0.883i)4-s + (−0.994 + 0.108i)5-s + (−0.796 − 0.605i)6-s + (−0.947 + 0.319i)7-s + (−0.0541 − 0.998i)8-s + (−0.725 + 0.687i)9-s + (−0.796 + 0.605i)10-s + (0.561 − 0.827i)11-s + (−0.994 − 0.108i)12-s + (0.725 + 0.687i)13-s + (−0.647 + 0.762i)14-s + (0.468 + 0.883i)15-s + (−0.561 − 0.827i)16-s + (−0.947 − 0.319i)17-s + ⋯
L(s,χ)  = 1  + (0.856 − 0.515i)2-s + (−0.370 − 0.928i)3-s + (0.468 − 0.883i)4-s + (−0.994 + 0.108i)5-s + (−0.796 − 0.605i)6-s + (−0.947 + 0.319i)7-s + (−0.0541 − 0.998i)8-s + (−0.725 + 0.687i)9-s + (−0.796 + 0.605i)10-s + (0.561 − 0.827i)11-s + (−0.994 − 0.108i)12-s + (0.725 + 0.687i)13-s + (−0.647 + 0.762i)14-s + (0.468 + 0.883i)15-s + (−0.561 − 0.827i)16-s + (−0.947 − 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.996 + 0.0870i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.996 + 0.0870i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $-0.996 + 0.0870i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (24, \cdot )$
Sato-Tate  :  $\mu(58)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (1:\ ),\ -0.996 + 0.0870i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.05320709956 - 1.220016541i$
$L(\frac12,\chi)$  $\approx$  $-0.05320709956 - 1.220016541i$
$L(\chi,1)$  $\approx$  0.7555154866 - 0.7764156431i
$L(1,\chi)$  $\approx$  0.7555154866 - 0.7764156431i

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

−32.96546656215547614267698171710, −32.075052739297602377726066391, −31.167230037125328518859246269252, −29.89985823332285735982812886338, −28.49254014422706602711346664762, −27.3143807036446479292410251112, −26.236056989673847580304402117596, −25.21231282544451125895734500429, −23.511271814040815886971415289832, −22.893291161398208481487308176716, −22.09161730219262839904107351480, −20.572319069688793351195199361365, −19.78761137910361074296657399809, −17.55757760942753948367372303114, −16.31809751938766247648976668767, −15.6563344680523095775909404204, −14.67742583569185161686544856233, −12.950754050843690035258997964524, −11.86572061501920559888324104853, −10.58106278192372097161041282213, −8.82505254370860459483493711014, −7.17738505888766371641094935769, −5.81014533173642734991821490061, −4.1988933892459122396153269300, −3.49266788821337171240072202580, 0.51695089609634023128322605945, 2.627549058471976591303923626475, 4.14423850013148393872833159822, 6.072459917101450137059585682677, 6.92808787012669584953447206826, 8.87488366561585410835558632599, 11.04508254837863150227416336660, 11.72713263090613639951072579677, 12.90583674973779254465738328079, 13.862233448702887393551496091135, 15.46397614268248362872031023081, 16.53559360263013156464829934214, 18.667507327033069889357964937539, 19.209576902026307505039289032798, 20.23581311165696318984103504669, 22.106350613430951995206998194694, 22.71918757729666956194394979051, 23.90074987209572320452522793970, 24.54849020485858332154269843084, 26.14683564290855041172520707962, 27.944992363413005563612052066803, 28.75298148092398479219446509477, 29.81182470211600454315379267989, 30.77263331368107093358198135942, 31.59208664281292257535904740486

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