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Properties

Label 1-61-61.3-r0-0-0
Degree $1$
Conductor $61$
Sign $0.615 + 0.788i$
Analytic cond. $0.283282$
Root an. cond. $0.283282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038414046 + 0.5064709511i\)
\(L(\frac12)\) \(\approx\) \(1.038414046 + 0.5064709511i\)
\(L(1)\) \(\approx\) \(1.201823380 + 0.3989345169i\)
\(L(1)\) \(\approx\) \(1.201823380 + 0.3989345169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 - T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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