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Properties

Label 1-61-61.18-r1-0-0
Degree $1$
Conductor $61$
Sign $0.183 - 0.982i$
Analytic cond. $6.55536$
Root an. cond. $6.55536$
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  + (0.207 + 0.978i)2-s + (−0.309 + 0.951i)3-s + (−0.913 + 0.406i)4-s + (0.104 − 0.994i)5-s + (−0.994 − 0.104i)6-s + (−0.743 − 0.669i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.994 − 0.104i)10-s + i·11-s + (−0.104 − 0.994i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.913 + 0.406i)15-s + (0.669 − 0.743i)16-s + (0.406 + 0.913i)17-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.309 + 0.951i)3-s + (−0.913 + 0.406i)4-s + (0.104 − 0.994i)5-s + (−0.994 − 0.104i)6-s + (−0.743 − 0.669i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.994 − 0.104i)10-s + i·11-s + (−0.104 − 0.994i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.913 + 0.406i)15-s + (0.669 − 0.743i)16-s + (0.406 + 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.183 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.183 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.183 - 0.982i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ 0.183 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1336185273 - 0.1109673565i\)
\(L(\frac12)\) \(\approx\) \(0.1336185273 - 0.1109673565i\)
\(L(1)\) \(\approx\) \(0.5701189762 + 0.3166997891i\)
\(L(1)\) \(\approx\) \(0.5701189762 + 0.3166997891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (0.207 - 0.978i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (-0.406 + 0.913i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (-0.207 - 0.978i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (0.994 - 0.104i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (0.406 - 0.913i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−31.96414445747699878740643991268, −31.26668540234283272898094106867, −29.988032622282161394796727099028, −29.407965346188397342681745248613, −28.55815212893030206198620051738, −27.11471260931550599586725103970, −25.88922499004758635162486173653, −24.48284466109770582040578824796, −23.17781144889241122956950400227, −22.35267927481512998219862666481, −21.44002977497455218493114240620, −19.61131149458073349552832749922, −18.776068539536242607097527608896, −18.301442537749800825288023093326, −16.6002598520464949586929966357, −14.537268010710074631177375643738, −13.66561489312877153099070439154, −12.33045885568320788010402310950, −11.47760113910957694555745179406, −10.18485018624091653433701574904, −8.63930040107530066391018488142, −6.75313290058799439705315937561, −5.59353861156279606267522083743, −3.27275789783716310717692570797, −2.10507786765962107569849757401, 0.0864100535052616952583504307, 3.79607469271373463366713328838, 4.85772001901585284866501234682, 6.07870741998025174211209683035, 7.78495160162771906424914971694, 9.29451141823276885172174721340, 10.14133861103178060774186731335, 12.33155544411206586622769633265, 13.32862396496721478882767489349, 14.982616658111124075646471501324, 15.79892754466055410219646623233, 17.05754136636647042636252120821, 17.39013560706993942128828744848, 19.68786929967967567317623019130, 20.84686778227909130008971624244, 22.126907579538458001421168472840, 23.05875448558269953940399863102, 24.04344451055218423759592886073, 25.54286247414737399607875742386, 26.188214679592200637030534403982, 27.64701834129467563750376046876, 28.18622520843217082398543034047, 29.81063028716239459840217261023, 31.52556514363732635195812374689, 32.44253934146727130344936197468

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