Properties

Degree 1
Conductor 61
Sign $0.468 + 0.883i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 + 0.951i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)10-s − 11-s + (−0.809 − 0.587i)12-s + 13-s + 14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s,χ)  = 1  + (−0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 + 0.951i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)10-s − 11-s + (−0.809 − 0.587i)12-s + 13-s + 14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.468 + 0.883i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.468 + 0.883i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $0.468 + 0.883i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (52, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 61,\ (0:\ ),\ 0.468 + 0.883i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5319145328 + 0.3198292745i$
$L(\frac12,\chi)$  $\approx$  $0.5319145328 + 0.3198292745i$
$L(\chi,1)$  $\approx$  0.7333310489 + 0.1391534797i
$L(1,\chi)$  $\approx$  0.7333310489 + 0.1391534797i

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

−32.30890056545871781503126567631, −31.42656477775427372087487580796, −30.35581189513347126276684826086, −28.84369537783007766588409476327, −27.84043536155551604419451288077, −26.37798945167706872508521909115, −25.7577505992264017077773236205, −24.34113102472579605550066944465, −23.55540409631196834178439752110, −23.04168814822989310758522182297, −20.64289623894446576554519925942, −19.5121445633790118198521065837, −18.61671474219844901956004006296, −17.32905818646157673220055865189, −16.24818154927051066841465808416, −15.07179079810271123104942807713, −13.53280847934440095018684044225, −12.906109699752999518931314114918, −10.97241916916508559168179048204, −9.12198456235563658279298951986, −7.89053542806036823598883784216, −7.21546161687899909362116170942, −5.59616616126942809200814553901, −3.76597233064210581214305193263, −0.894856830738225738813380516552, 2.74372540005968642897677536023, 3.65354993653957266841593050714, 5.32072472207840260975527494734, 7.89340844170291311213611951352, 8.96910806941764074533485943773, 10.29984703866464687792044243146, 11.22111646487297622222452787713, 12.47061431019101522134759629057, 14.11877211673066897214024063967, 15.461219508261693696854733711, 16.424137593960491276565354960940, 18.42092003561823862495318820030, 18.96533493327188960202527289986, 20.445959225766302925690632869883, 21.18223068709545278200420310068, 22.43200281879284679823106882976, 23.14794780375329573051739394763, 25.464982473549101209127470517349, 26.32603598406636196350427397257, 27.361806362484143002937418573246, 28.12063271455347964131795947305, 29.20833104584376364227291555363, 30.9123851689057735464382602500, 31.28430346350649603643634394460, 32.333784650387129858746133655353

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