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Properties

Label 1-61-61.12-r0-0-0
Degree $1$
Conductor $61$
Sign $0.962 + 0.270i$
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  + (0.669 + 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 + 0.994i)4-s + (0.913 − 0.406i)5-s + (0.913 − 0.406i)6-s + (−0.978 − 0.207i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.913 + 0.406i)10-s + 11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)17-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 + 0.994i)4-s + (0.913 − 0.406i)5-s + (0.913 − 0.406i)6-s + (−0.978 − 0.207i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.913 + 0.406i)10-s + 11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (−0.104 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.962 + 0.270i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.322693065 + 0.1824834269i\)
\(L(\frac12)\) \(\approx\) \(1.322693065 + 0.1824834269i\)
\(L(1)\) \(\approx\) \(1.430792096 + 0.1819230454i\)
\(L(1)\) \(\approx\) \(1.430792096 + 0.1819230454i\)

Euler product

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

−32.35629395035699339213198570380, −31.69210462522182605107341766617, −30.11735352705525191492315191749, −29.37724076545146552730253568809, −28.16219044407554605775243812062, −27.16515341742261067728078370957, −25.709559417117719228408597599583, −24.84933249659629794030578331561, −22.795489388035498777773342719372, −22.2191618372792069612133848213, −21.39714796074035587166258120270, −20.13138003904431532991296461108, −19.27253866811969751900863281074, −17.62838155941287909163139258147, −16.03222548894384626652312981695, −14.78558925994019318559678847361, −13.90725567728124763861618227205, −12.62269269401520193021931964159, −11.023725604094676868604548366607, −9.877997229858242916979110674325, −9.23256734770494618073151566850, −6.4405469027192969994555300431, −5.208702132746276048466085223829, −3.57859383457997580860638934962, −2.46273271448130016417989021119, 2.211549375892348612868539608438, 4.057495415858771086653725497536, 6.15301111235568168505350274966, 6.61228097042314439867375709100, 8.33931292431334942744877995620, 9.53559184826933572078387697777, 12.01168181285223177270053498750, 12.95558410424600808597569449811, 13.81940249998491266663295594695, 14.85022049804415971928536227216, 16.769992976853064579755938221382, 17.213017551575537329373541764294, 18.83985556603394817949100825782, 20.10842488793530827540250784936, 21.55351117218188205353437110724, 22.59484519971203850650848808066, 23.88003218005248273061038739677, 24.706543763502428143141375489603, 25.64210335802645789767714211511, 26.35873426289461623461953219660, 28.48397660318353154033108655752, 29.66197689867740134189627943231, 30.29468282001178223599548466210, 31.80939837080270424711974087823, 32.391863970190405267605841522390

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