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Properties

Label 1-61-61.15-r0-0-0
Degree $1$
Conductor $61$
Sign $0.811 - 0.583i$
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  + (−0.978 + 0.207i)2-s + (0.309 − 0.951i)3-s + (0.913 − 0.406i)4-s + (−0.104 + 0.994i)5-s + (−0.104 + 0.994i)6-s + (0.669 − 0.743i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.104 − 0.994i)10-s + 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 + (0.913 − 0.406i)17-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.309 − 0.951i)3-s + (0.913 − 0.406i)4-s + (−0.104 + 0.994i)5-s + (−0.104 + 0.994i)6-s + (0.669 − 0.743i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.104 − 0.994i)10-s + 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 + (0.913 − 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.811 - 0.583i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.811 - 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6576641991 - 0.2118498124i\)
\(L(\frac12)\) \(\approx\) \(0.6576641991 - 0.2118498124i\)
\(L(1)\) \(\approx\) \(0.7693410903 - 0.1443957532i\)
\(L(1)\) \(\approx\) \(0.7693410903 - 0.1443957532i\)

Euler product

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

−32.71449617597021380285786615795, −31.59679516112480510702745239965, −30.45218662162139620742903726286, −28.82118177884867049896897038617, −27.90854304286348323206052336550, −27.434769389567527297640436509915, −26.14871237200947019135414043052, −25.02608351651229946873894981023, −24.14199608715014313172608155324, −21.90034738275350880507321296925, −21.18229980120459524239405782226, −20.10989321729203366852465208489, −19.210437015558351735992777259600, −17.502217901305443463340839686293, −16.61681985741893199857995548118, −15.61006676060443348875340518200, −14.34350824493810111293587679476, −12.157680581575342546479958738050, −11.32823587206311850385135100901, −9.54650773557755307852635445595, −9.01719226282563221725639832093, −7.79178227605321916719508410325, −5.553343962396524515597596434069, −3.92633546197191467504414305820, −1.917808577495930102907404752321, 1.42653028916977855999690139236, 3.13163930460288100374367342182, 6.05161805154149258356718366276, 7.34215268052553503820280752208, 7.93034079493491508097345903408, 9.69904846085493654530910509798, 11.02765268911246545600857911211, 12.1375134944414794352624965918, 14.24129182710326328894884125977, 14.7126118452175505023973634481, 16.66696926579213584495238828898, 17.8136842785197207752055594611, 18.521208313471407875342817095765, 19.71095073606827330436439299619, 20.500323084779592887176666874771, 22.536792069721546142008598043715, 23.77273904804065702435823456563, 24.819344386885024546775759183945, 25.732966479717879561063193157522, 26.86784073540988289910901809066, 27.67002455103582002869937083686, 29.55640791056216231554658796813, 29.84104459183109943717282904521, 30.89143809980600111616141026452, 32.68102855170979692198956942476

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