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Properties

Label 1-61-61.54-r1-0-0
Degree $1$
Conductor $61$
Sign $0.852 - 0.522i$
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  + (−0.406 + 0.913i)2-s + (0.809 − 0.587i)3-s + (−0.669 − 0.743i)4-s + (0.978 − 0.207i)5-s + (0.207 + 0.978i)6-s + (−0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.207 + 0.978i)10-s − i·11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (0.743 − 0.669i)17-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (0.809 − 0.587i)3-s + (−0.669 − 0.743i)4-s + (0.978 − 0.207i)5-s + (0.207 + 0.978i)6-s + (−0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.207 + 0.978i)10-s − i·11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (0.743 − 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.553568579 - 0.4383789294i\)
\(L(\frac12)\) \(\approx\) \(1.553568579 - 0.4383789294i\)
\(L(1)\) \(\approx\) \(1.168093842 - 0.03869793043i\)
\(L(1)\) \(\approx\) \(1.168093842 - 0.03869793043i\)

Euler product

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

−32.275027994122609050660882462481, −31.11403176109435171890224182513, −30.15014511225175652005985060743, −28.90862318206431413973857726210, −28.14546555822826886480261893390, −26.61687295324147269393766812338, −25.94300229583872365561178177439, −25.13659303739807374770439060349, −22.84619845841862127215311614479, −21.74798735978410394064724476681, −21.10211217284586049867177891036, −19.788092529236933942908998311711, −19.02378443491947703151742833059, −17.56838435798978316547260653623, −16.39618717822100481224642699056, −14.70307351046338554726427450538, −13.51359557192934559047427852593, −12.48040994721285156032769941391, −10.54424752381051717449669404293, −9.68393335568170588751763461453, −8.96109028953077210007965963961, −7.06866792601538805098511863009, −4.774520099798563289474447702493, −3.16442963148636069447097621259, −2.01788389520198598888401512404, 0.971924203687201451721885382660, 3.10421470660587801885403598936, 5.5413127685492987352751343072, 6.648061500081994815058153538961, 7.99973041262547441656282626023, 9.25638768289382775375789129142, 10.1089930186871471963461404542, 12.75577130258191238306884757348, 13.64752969795114733291980844130, 14.59444237284228970148313375817, 16.07965635036583790427192136216, 17.18302748290964212565616320068, 18.46452955082966075395335497768, 19.26456136231218639596176105826, 20.60604924890607133443867789354, 22.205151713515060475243889835317, 23.5001007393978708155622784463, 24.895445866749669367315572326870, 25.210659960924360807718004623292, 26.28772568214988010521647560898, 27.34749737849005089657816859350, 29.16306121984120873018004153057, 29.57877805907529445608742116077, 31.55430952071167923902863609863, 32.25474881501169931311642339483

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