Properties

Label 1-571-571.206-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.583 + 0.812i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (−0.0825 + 0.996i)2-s + (0.997 − 0.0660i)3-s + (−0.986 − 0.164i)4-s + (−0.934 − 0.355i)5-s + (−0.0165 + 0.999i)6-s + (−0.909 − 0.416i)7-s + (0.245 − 0.969i)8-s + (0.991 − 0.131i)9-s + (0.431 − 0.901i)10-s + (0.431 + 0.901i)11-s + (−0.995 − 0.0990i)12-s + (−0.277 + 0.960i)13-s + (0.490 − 0.871i)14-s + (−0.956 − 0.293i)15-s + (0.945 + 0.324i)16-s + (−0.574 − 0.818i)17-s + ⋯
L(s)  = 1  + (−0.0825 + 0.996i)2-s + (0.997 − 0.0660i)3-s + (−0.986 − 0.164i)4-s + (−0.934 − 0.355i)5-s + (−0.0165 + 0.999i)6-s + (−0.909 − 0.416i)7-s + (0.245 − 0.969i)8-s + (0.991 − 0.131i)9-s + (0.431 − 0.901i)10-s + (0.431 + 0.901i)11-s + (−0.995 − 0.0990i)12-s + (−0.277 + 0.960i)13-s + (0.490 − 0.871i)14-s + (−0.956 − 0.293i)15-s + (0.945 + 0.324i)16-s + (−0.574 − 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.583 + 0.812i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.583 + 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4903251701 + 0.9553468089i\)
\(L(\frac12)\) \(\approx\) \(0.4903251701 + 0.9553468089i\)
\(L(1)\) \(\approx\) \(0.8363203276 + 0.5030653778i\)
\(L(1)\) \(\approx\) \(0.8363203276 + 0.5030653778i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.0825 + 0.996i)T \)
3 \( 1 + (0.997 - 0.0660i)T \)
5 \( 1 + (-0.934 - 0.355i)T \)
7 \( 1 + (-0.909 - 0.416i)T \)
11 \( 1 + (0.431 + 0.901i)T \)
13 \( 1 + (-0.277 + 0.960i)T \)
17 \( 1 + (-0.574 - 0.818i)T \)
19 \( 1 + (0.371 + 0.928i)T \)
23 \( 1 + (-0.846 + 0.533i)T \)
29 \( 1 + (0.789 - 0.614i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (-0.999 - 0.0330i)T \)
41 \( 1 + (0.546 + 0.837i)T \)
43 \( 1 + (0.180 + 0.983i)T \)
47 \( 1 + (0.945 + 0.324i)T \)
53 \( 1 + (-0.518 + 0.854i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.973 + 0.229i)T \)
67 \( 1 + (-0.518 + 0.854i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.999 - 0.0330i)T \)
79 \( 1 + (-0.461 - 0.887i)T \)
83 \( 1 + (-0.0165 + 0.999i)T \)
89 \( 1 + (-0.0165 + 0.999i)T \)
97 \( 1 + (0.894 - 0.446i)T \)
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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

−22.41886992055258181448955116152, −22.22607135358840937885689952067, −21.257724381590541231149723476046, −20.00854739479214560102935306692, −19.79811694145665830151959857583, −19.06824670202757092856154011033, −18.450049861144342940374703663230, −17.298408378162122453797196731292, −15.901178676937531412247118884186, −15.4030980846042018059443848905, −14.30846772653363551153555648095, −13.54308140558192952703000894292, −12.60788668203303584815048928666, −11.99602080497023374677776153564, −10.74913288867935188505646666955, −10.173306165449901316562049772335, −8.9372684701073479575441381790, −8.534895973806386060129922051981, −7.51471586573244912443448753423, −6.2670396711989964419184453111, −4.72222527732247905847106795242, −3.63889190453909430054085876963, −3.15775260846615474967277506013, −2.273688712842784370240353285117, −0.56107475665106585754010036822, 1.30255674315817853164290268406, 3.02288963015621408822596491082, 4.2234942495479640083077658042, 4.46140001856605169848362345718, 6.26747935772158712955664467688, 7.18870235010629406400190780372, 7.64100074147169835841742326160, 8.73131368183223520714018269625, 9.47828230420803297681248549232, 10.10514249160205861450643303124, 11.94477142465306669539593793992, 12.61662369847093149005365836926, 13.72782547184052630616973477811, 14.20424317076206156687213117046, 15.30318328323779408364824572344, 15.883612544530642228189276264072, 16.495458669308900616580930493943, 17.59897377078203895078535056966, 18.670625168579709478820949810074, 19.429547646911465750151999435398, 19.90065645286016519751615665389, 20.905210482857730468920149991317, 22.15061218149277314533820522494, 22.9845401674364128819130163677, 23.57413681611845716621948282092

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