Properties

Degree 1
Conductor 61
Sign $0.811 + 0.583i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $0.811 + 0.583i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (57, \cdot )$
Sato-Tate  :  $\mu(15)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 61,\ (0:\ ),\ 0.811 + 0.583i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6576641991 + 0.2118498124i$
$L(\frac12,\chi)$  $\approx$  $0.6576641991 + 0.2118498124i$
$L(\chi,1)$  $\approx$  0.7693410903 + 0.1443957532i
$L(1,\chi)$  $\approx$  0.7693410903 + 0.1443957532i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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