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Properties

Label 1-61-61.27-r0-0-0
Degree $1$
Conductor $61$
Sign $0.468 - 0.883i$
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  + (−0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)10-s − 11-s + (−0.809 + 0.587i)12-s + 13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)10-s − 11-s + (−0.809 + 0.587i)12-s + 13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.468 - 0.883i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.468 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5319145328 - 0.3198292745i\)
\(L(\frac12)\) \(\approx\) \(0.5319145328 - 0.3198292745i\)
\(L(1)\) \(\approx\) \(0.7333310489 - 0.1391534797i\)
\(L(1)\) \(\approx\) \(0.7333310489 - 0.1391534797i\)

Euler product

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

−32.333784650387129858746133655353, −31.28430346350649603643634394460, −30.9123851689057735464382602500, −29.20833104584376364227291555363, −28.12063271455347964131795947305, −27.361806362484143002937418573246, −26.32603598406636196350427397257, −25.464982473549101209127470517349, −23.14794780375329573051739394763, −22.43200281879284679823106882976, −21.18223068709545278200420310068, −20.445959225766302925690632869883, −18.96533493327188960202527289986, −18.42092003561823862495318820030, −16.424137593960491276565354960940, −15.461219508261693696854733711, −14.11877211673066897214024063967, −12.47061431019101522134759629057, −11.22111646487297622222452787713, −10.29984703866464687792044243146, −8.96910806941764074533485943773, −7.89340844170291311213611951352, −5.32072472207840260975527494734, −3.65354993653957266841593050714, −2.74372540005968642897677536023, 0.894856830738225738813380516552, 3.76597233064210581214305193263, 5.59616616126942809200814553901, 7.21546161687899909362116170942, 7.89053542806036823598883784216, 9.12198456235563658279298951986, 10.97241916916508559168179048204, 12.906109699752999518931314114918, 13.53280847934440095018684044225, 15.07179079810271123104942807713, 16.24818154927051066841465808416, 17.32905818646157673220055865189, 18.61671474219844901956004006296, 19.5121445633790118198521065837, 20.64289623894446576554519925942, 23.04168814822989310758522182297, 23.55540409631196834178439752110, 24.34113102472579605550066944465, 25.7577505992264017077773236205, 26.37798945167706872508521909115, 27.84043536155551604419451288077, 28.84369537783007766588409476327, 30.35581189513347126276684826086, 31.42656477775427372087487580796, 32.30890056545871781503126567631

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