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Properties

Label 1-571-571.104-r1-0-0
Degree $1$
Conductor $571$
Sign $0.261 + 0.965i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (0.592 − 0.805i)2-s + (−0.996 + 0.0880i)3-s + (−0.298 − 0.954i)4-s + (0.884 + 0.466i)5-s + (−0.518 + 0.854i)6-s + (−0.0495 + 0.998i)7-s + (−0.945 − 0.324i)8-s + (0.984 − 0.175i)9-s + (0.899 − 0.436i)10-s + (0.0715 + 0.997i)11-s + (0.381 + 0.924i)12-s + (0.930 + 0.366i)13-s + (0.775 + 0.631i)14-s + (−0.922 − 0.386i)15-s + (−0.821 + 0.569i)16-s + (−0.287 − 0.957i)17-s + ⋯
L(s)  = 1  + (0.592 − 0.805i)2-s + (−0.996 + 0.0880i)3-s + (−0.298 − 0.954i)4-s + (0.884 + 0.466i)5-s + (−0.518 + 0.854i)6-s + (−0.0495 + 0.998i)7-s + (−0.945 − 0.324i)8-s + (0.984 − 0.175i)9-s + (0.899 − 0.436i)10-s + (0.0715 + 0.997i)11-s + (0.381 + 0.924i)12-s + (0.930 + 0.366i)13-s + (0.775 + 0.631i)14-s + (−0.922 − 0.386i)15-s + (−0.821 + 0.569i)16-s + (−0.287 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.261 + 0.965i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.261 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.206704236 + 0.9233391895i\)
\(L(\frac12)\) \(\approx\) \(1.206704236 + 0.9233391895i\)
\(L(1)\) \(\approx\) \(1.112238935 - 0.1119650074i\)
\(L(1)\) \(\approx\) \(1.112238935 - 0.1119650074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.592 - 0.805i)T \)
3 \( 1 + (-0.996 + 0.0880i)T \)
5 \( 1 + (0.884 + 0.466i)T \)
7 \( 1 + (-0.0495 + 0.998i)T \)
11 \( 1 + (0.0715 + 0.997i)T \)
13 \( 1 + (0.930 + 0.366i)T \)
17 \( 1 + (-0.287 - 0.957i)T \)
19 \( 1 + (0.857 + 0.514i)T \)
23 \( 1 + (-0.956 - 0.293i)T \)
29 \( 1 + (0.350 + 0.936i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (-0.537 + 0.843i)T \)
41 \( 1 + (-0.716 + 0.697i)T \)
43 \( 1 + (-0.693 - 0.720i)T \)
47 \( 1 + (-0.904 - 0.426i)T \)
53 \( 1 + (0.949 - 0.314i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (-0.739 - 0.673i)T \)
67 \( 1 + (-0.202 + 0.979i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.999 - 0.0440i)T \)
79 \( 1 + (0.917 + 0.396i)T \)
83 \( 1 + (-0.480 - 0.876i)T \)
89 \( 1 + (-0.999 - 0.0220i)T \)
97 \( 1 + (0.815 - 0.578i)T \)
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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

−22.97687348518339371384079248915, −22.147406862698547134126645203032, −21.45831795103980272880813561900, −20.79602629351254644113315305834, −19.57430491733396057376234430787, −18.06429467767195706688939932085, −17.72336137410767023085624167950, −16.85169582656698764781364038701, −16.27161073517808094719349163057, −15.5883549852137713244127782671, −14.08302431200128411482689857684, −13.52262403561080985916952336228, −12.96814189975976210022096861686, −11.89233410875098057091932507054, −10.899771424960914432390199488624, −10.03659709705579935872119883221, −8.78569191689075504172651100372, −7.84293356246723831437867059762, −6.67724288248696338653952340151, −6.05028505069756001011262070061, −5.35193895803292136930563948950, −4.325980273483084439190375987695, −3.38542602012342130478168878965, −1.50299488202043486685584740359, −0.34945693634991465599451039288, 1.38130402218691680919318481911, 2.153272061160411340354123674043, 3.37401545532266022272232864947, 4.67257997016672831851417914271, 5.43897909212385238265561943542, 6.18732006360702747873514640478, 6.967245788958276891533657816709, 8.87674375769119832643872366747, 9.84935379632021973541697812971, 10.25643722260243719291456022151, 11.57374318089442563814502448346, 11.83034194188542126506268477148, 12.91269136905459821123337361585, 13.665520753833778026829749676354, 14.68666437718533432075680681715, 15.52490185081809737622352781055, 16.41461108812732367803409078043, 17.79322679173963962043279020624, 18.32580013102120979816579694790, 18.65402625434341492164837842351, 20.21296186363484447517405780130, 20.92413064493665156171304105945, 21.72476506109229099325266080349, 22.40621204229214381530583874695, 22.76043250040965633639902495889

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