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Properties

Label 1-571-571.245-r0-0-0
Degree $1$
Conductor $571$
Sign $0.397 + 0.917i$
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  + (0.137 + 0.990i)2-s + (−0.949 − 0.314i)3-s + (−0.962 + 0.272i)4-s + (−0.982 + 0.186i)5-s + (0.180 − 0.983i)6-s + (−0.0165 + 0.999i)7-s + (−0.401 − 0.915i)8-s + (0.802 + 0.596i)9-s + (−0.319 − 0.947i)10-s + (−0.660 − 0.750i)11-s + (0.999 + 0.0440i)12-s + (0.840 − 0.542i)13-s + (−0.992 + 0.120i)14-s + (0.991 + 0.131i)15-s + (0.851 − 0.523i)16-s + (−0.997 − 0.0770i)17-s + ⋯
L(s)  = 1  + (0.137 + 0.990i)2-s + (−0.949 − 0.314i)3-s + (−0.962 + 0.272i)4-s + (−0.982 + 0.186i)5-s + (0.180 − 0.983i)6-s + (−0.0165 + 0.999i)7-s + (−0.401 − 0.915i)8-s + (0.802 + 0.596i)9-s + (−0.319 − 0.947i)10-s + (−0.660 − 0.750i)11-s + (0.999 + 0.0440i)12-s + (0.840 − 0.542i)13-s + (−0.992 + 0.120i)14-s + (0.991 + 0.131i)15-s + (0.851 − 0.523i)16-s + (−0.997 − 0.0770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.397 + 0.917i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.397 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4956430794 + 0.3254630051i\)
\(L(\frac12)\) \(\approx\) \(0.4956430794 + 0.3254630051i\)
\(L(1)\) \(\approx\) \(0.5475466254 + 0.2688213283i\)
\(L(1)\) \(\approx\) \(0.5475466254 + 0.2688213283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.137 + 0.990i)T \)
3 \( 1 + (-0.949 - 0.314i)T \)
5 \( 1 + (-0.982 + 0.186i)T \)
7 \( 1 + (-0.0165 + 0.999i)T \)
11 \( 1 + (-0.660 - 0.750i)T \)
13 \( 1 + (0.840 - 0.542i)T \)
17 \( 1 + (-0.997 - 0.0770i)T \)
19 \( 1 + (0.00551 - 0.999i)T \)
23 \( 1 + (-0.995 - 0.0990i)T \)
29 \( 1 + (0.451 + 0.892i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.775 - 0.631i)T \)
41 \( 1 + (0.904 + 0.426i)T \)
43 \( 1 + (0.815 + 0.578i)T \)
47 \( 1 + (0.0275 + 0.999i)T \)
53 \( 1 + (-0.693 + 0.720i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (0.0715 - 0.997i)T \)
67 \( 1 + (0.970 + 0.240i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.159 + 0.987i)T \)
79 \( 1 + (0.884 + 0.466i)T \)
83 \( 1 + (-0.942 + 0.335i)T \)
89 \( 1 + (0.761 + 0.648i)T \)
97 \( 1 + (0.618 - 0.785i)T \)
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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

−22.98148792406987124417256199881, −22.46225058034942772947726185692, −21.31188814438702198031635034999, −20.56747898624188087953303025927, −20.020262926820975371182610429264, −18.96164971432239971847118161422, −18.138244586394095490149661017552, −17.40682357176122274164098627026, −16.35560071344484484718320033736, −15.68478637054951762728821342754, −14.57745180659647669973169998662, −13.430549570866883026158879559550, −12.655528311646721578949091364827, −11.841596163384631350842201040584, −11.09704489682305225337075403562, −10.46313896456370610702729813715, −9.64987027980689825478170756096, −8.396909165377240066887757463996, −7.3820158667340748649791563584, −6.20029204920199259847237608955, −4.94999712577606167074064027767, −4.11798584422574815763490057647, −3.741152345027045092283515989826, −1.92121716379685270453124956135, −0.65157102182880221639851684373, 0.62601036062938525730592394456, 2.7163077201660976735837208758, 4.029457984979566099807744375, 5.00662315481918423701680295876, 5.88006018712347789169579061747, 6.551965929913474053498296179622, 7.66156127402267984810875500048, 8.31767426561843507727145266386, 9.28725377277903618178486560778, 10.86238684280355453945266781659, 11.34164372160417301872857858580, 12.60833615683448938457739207400, 12.983495613208318107459993820333, 14.2010161295877766546233633110, 15.42192132691823029442824453481, 15.863240557659715909534900984614, 16.31583211562697377905592094649, 17.72494974297726262540265577058, 18.19320295707209099034555161613, 18.81517902379640271152055987589, 19.83121769873206058853441706150, 21.3782169673705976411844621028, 22.11192041453172581656081322532, 22.64793271464290991851399337336, 23.65855970475481874524824283846

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