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Properties

Label 1-61-61.43-r1-0-0
Degree $1$
Conductor $61$
Sign $-0.593 - 0.805i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.593 - 0.805i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ -0.593 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4160910295 - 0.8232991914i\)
\(L(\frac12)\) \(\approx\) \(0.4160910295 - 0.8232991914i\)
\(L(1)\) \(\approx\) \(0.6977853172 - 0.3834176399i\)
\(L(1)\) \(\approx\) \(0.6977853172 - 0.3834176399i\)

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

−33.26290924432525070139074661139, −31.29960968384639535443111429379, −30.61860605662424684021035713996, −29.42767307525209861383284855359, −28.07666792469384274880838602037, −26.831006442665463452600924392545, −25.84810114361521294896892129693, −24.83644545764598798670031450871, −23.57947628263421159734756196558, −23.109609305774035167359595511452, −21.732674767174683215863126984098, −19.645348219729659210151457537235, −18.6341628010000398262988307381, −17.584904297746724421758446870832, −16.97277571662614443909098765922, −14.991433684312216139845135089547, −14.25452631688202540697741357971, −13.049481708649974425354235112507, −11.34075568534466311891431699920, −9.97301141788892250674979588865, −8.04125561723842088772444268975, −7.14679600025967477941577991295, −6.20549411624648660321056130018, −4.44952667376677574917979550392, −1.76883384343816511473956229830, 0.5686927018013226126238307616, 2.779380037325292101233392428040, 4.589560660260197685303217032066, 5.37885890638486568741665555794, 8.46945467544838344541098757115, 9.09019308450822765486559763227, 10.58417993242778773207947057477, 11.59911221207872829181982135773, 12.72905534489779892106180455982, 14.27794660638862604959724941991, 15.828184647842025909712859709, 17.07888898041005508722792348942, 18.00275470538554133206528103772, 19.660586962853932059289442583930, 20.71641341819028127287644753143, 21.46056500872708915123317660005, 22.35810223145156880333513777141, 23.86607798918160149592717065774, 25.26908750723511634888540890814, 27.05307672096397155217721819460, 27.40999752502514683333379509280, 28.54689223620447542902188614761, 29.26805483737258760266834811100, 30.84071586928920841238005109869, 32.02324520373213478690873673992

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