Properties

Degree 1
Conductor 61
Sign $0.615 + 0.788i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.615 + 0.788i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.615 + 0.788i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $0.615 + 0.788i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (3, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 61,\ (0:\ ),\ 0.615 + 0.788i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.038414046 + 0.5064709511i$
$L(\frac12,\chi)$  $\approx$  $1.038414046 + 0.5064709511i$
$L(\chi,1)$  $\approx$  1.201823380 + 0.3989345169i
$L(1,\chi)$  $\approx$  1.201823380 + 0.3989345169i

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

−32.283022354024017512829330193704, −31.46495741925934247410885338775, −30.14854377967546192442884920313, −28.78015530517898438990783632840, −28.305476781587001828322021596289, −27.40311057855537517726502363645, −25.45681300967089714043297172717, −23.93927025629964313722055552110, −23.52788681130979711237396537429, −21.929682119588115770915709067045, −21.13571234996188323438977269066, −20.53779761940775878156599641703, −18.66286297269784826374934793894, −17.41931872607565608913811305335, −15.94836749788909702974166951650, −15.08385168521392434531492638599, −13.37391429692839791747692296824, −12.3329587754986332128036324509, −11.216364709716284336914508425830, −10.13756118554743018428179710547, −8.58328441801921064454982486531, −5.9780900645171621425196428568, −5.21599810193109551820724862046, −4.0020856244533515215518277424, −1.69464310119038800479565744217, 2.43331487318047986331407999569, 4.50535948697483621866162453900, 5.88106775131665601477666933114, 6.97430033268576008991681664325, 8.012961094063363339548054906434, 10.72579506669883991975439679056, 11.42798436816465704373673827697, 13.119153954926037737233195202423, 13.864752407949856530390961556948, 15.239866992657960803845582022034, 16.56581765594190932011440886265, 17.78479542170748125481567842743, 18.47641111996590345273314386907, 20.611100520625794520197342428092, 21.7240419523795564492318252760, 22.89772625526265715471077647774, 23.52448839107190130874047080551, 24.58336085897407230291291238615, 25.84779042033117656347911389983, 26.86789018068306047151769820733, 28.51088566514445115261287776593, 29.93581382573327585225238758938, 30.26972477608462024958663763145, 31.431523830655657742823829174408, 33.16110567813610860292747651308

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