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Properties

Label 1-571-571.37-r0-0-0
Degree $1$
Conductor $571$
Sign $0.218 + 0.975i$
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  + (−0.926 + 0.376i)2-s + (−0.868 − 0.495i)3-s + (0.716 − 0.697i)4-s + (0.287 − 0.957i)5-s + (0.991 + 0.131i)6-s + (−0.956 − 0.293i)7-s + (−0.401 + 0.915i)8-s + (0.509 + 0.860i)9-s + (0.0935 + 0.995i)10-s + (0.815 + 0.578i)11-s + (−0.968 + 0.250i)12-s + (0.988 + 0.153i)13-s + (0.996 − 0.0880i)14-s + (−0.724 + 0.689i)15-s + (0.0275 − 0.999i)16-s + (−0.942 − 0.335i)17-s + ⋯
L(s)  = 1  + (−0.926 + 0.376i)2-s + (−0.868 − 0.495i)3-s + (0.716 − 0.697i)4-s + (0.287 − 0.957i)5-s + (0.991 + 0.131i)6-s + (−0.956 − 0.293i)7-s + (−0.401 + 0.915i)8-s + (0.509 + 0.860i)9-s + (0.0935 + 0.995i)10-s + (0.815 + 0.578i)11-s + (−0.968 + 0.250i)12-s + (0.988 + 0.153i)13-s + (0.996 − 0.0880i)14-s + (−0.724 + 0.689i)15-s + (0.0275 − 0.999i)16-s + (−0.942 − 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.218 + 0.975i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.218 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2912782329 + 0.2332312062i\)
\(L(\frac12)\) \(\approx\) \(0.2912782329 + 0.2332312062i\)
\(L(1)\) \(\approx\) \(0.4860826683 + 0.0004490893570i\)
\(L(1)\) \(\approx\) \(0.4860826683 + 0.0004490893570i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.926 + 0.376i)T \)
3 \( 1 + (-0.868 - 0.495i)T \)
5 \( 1 + (0.287 - 0.957i)T \)
7 \( 1 + (-0.956 - 0.293i)T \)
11 \( 1 + (0.815 + 0.578i)T \)
13 \( 1 + (0.988 + 0.153i)T \)
17 \( 1 + (-0.942 - 0.335i)T \)
19 \( 1 + (0.411 + 0.911i)T \)
23 \( 1 + (-0.213 + 0.976i)T \)
29 \( 1 + (-0.998 + 0.0550i)T \)
31 \( 1 + (-0.879 + 0.475i)T \)
37 \( 1 + (-0.709 + 0.705i)T \)
41 \( 1 + (-0.821 - 0.569i)T \)
43 \( 1 + (-0.917 + 0.396i)T \)
47 \( 1 + (0.851 + 0.523i)T \)
53 \( 1 + (-0.644 + 0.764i)T \)
59 \( 1 + (0.945 - 0.324i)T \)
61 \( 1 + (-0.693 + 0.720i)T \)
67 \( 1 + (0.984 + 0.175i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.256 - 0.966i)T \)
79 \( 1 + (0.938 + 0.345i)T \)
83 \( 1 + (-0.381 - 0.924i)T \)
89 \( 1 + (-0.609 + 0.792i)T \)
97 \( 1 + (0.884 + 0.466i)T \)
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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

−22.619744485818599177489875209378, −22.1440506314747037584918189805, −21.6549815340356339942692535744, −20.50424642362880709440664432037, −19.62059424556776767836720042156, −18.607412785732949088924002301855, −18.23229934621429505283016935872, −17.240968169129598041220258999371, −16.51878302363800406392264582743, −15.70362681515566469802496299055, −15.04212601942356561249500019782, −13.52528372081388347877542585504, −12.62062135733809223275914761256, −11.443330797017238124569089307566, −11.061691195009679214506252470128, −10.21451674267909410479817336334, −9.35979114940104856013440550268, −8.675210467392136770191730165534, −6.983120076774497890488641179216, −6.51968349095382636274688733154, −5.745789932128669062208545090376, −3.87020838945194329408504321735, −3.27413277753046969046942523443, −1.93873680727329074551408988082, −0.32279477818591810802649302121, 1.18482389630483586902538252093, 1.84731561524358298243997017088, 3.80888205207264799988124419597, 5.1785182351385227415963715729, 6.03226985262639618344584875018, 6.74275982618040903559440899679, 7.60147190777947765728997930907, 8.813688653744339346181071683272, 9.51791586603485285260394200912, 10.3708486563265200234753360236, 11.4449958624488868480509573657, 12.18695443411903530025970043624, 13.17074284880623394820574443620, 13.948382179113383543256671894149, 15.53065821959139808374620708210, 16.161266187394114194740179387, 16.83318940316425966477093223543, 17.43305088198351364633482421323, 18.28101890863212470861543415483, 19.08441848805890113589792625690, 20.02979609893678072756164685613, 20.543156461934173870046733452975, 21.92027219497867902087976975139, 22.818613709432048342456507187237, 23.620744306459195394418803416150

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