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Properties

Label 1-571-571.115-r0-0-0
Degree $1$
Conductor $571$
Sign $0.181 + 0.983i$
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  + (−0.298 − 0.954i)2-s + (0.471 + 0.882i)3-s + (−0.821 + 0.569i)4-s + (−0.942 − 0.335i)5-s + (0.701 − 0.712i)6-s + (−0.213 − 0.976i)7-s + (0.789 + 0.614i)8-s + (−0.556 + 0.831i)9-s + (−0.0385 + 0.999i)10-s + (0.884 + 0.466i)11-s + (−0.889 − 0.456i)12-s + (−0.992 − 0.120i)13-s + (−0.868 + 0.495i)14-s + (−0.148 − 0.988i)15-s + (0.350 − 0.936i)16-s + (0.999 + 0.0440i)17-s + ⋯
L(s)  = 1  + (−0.298 − 0.954i)2-s + (0.471 + 0.882i)3-s + (−0.821 + 0.569i)4-s + (−0.942 − 0.335i)5-s + (0.701 − 0.712i)6-s + (−0.213 − 0.976i)7-s + (0.789 + 0.614i)8-s + (−0.556 + 0.831i)9-s + (−0.0385 + 0.999i)10-s + (0.884 + 0.466i)11-s + (−0.889 − 0.456i)12-s + (−0.992 − 0.120i)13-s + (−0.868 + 0.495i)14-s + (−0.148 − 0.988i)15-s + (0.350 − 0.936i)16-s + (0.999 + 0.0440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4275512108 + 0.3559925120i\)
\(L(\frac12)\) \(\approx\) \(0.4275512108 + 0.3559925120i\)
\(L(1)\) \(\approx\) \(0.7002379940 - 0.05325413971i\)
\(L(1)\) \(\approx\) \(0.7002379940 - 0.05325413971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.298 - 0.954i)T \)
3 \( 1 + (0.471 + 0.882i)T \)
5 \( 1 + (-0.942 - 0.335i)T \)
7 \( 1 + (-0.213 - 0.976i)T \)
11 \( 1 + (0.884 + 0.466i)T \)
13 \( 1 + (-0.992 - 0.120i)T \)
17 \( 1 + (0.999 + 0.0440i)T \)
19 \( 1 + (-0.899 - 0.436i)T \)
23 \( 1 + (-0.277 + 0.960i)T \)
29 \( 1 + (-0.754 + 0.656i)T \)
31 \( 1 + (-0.986 + 0.164i)T \)
37 \( 1 + (0.874 + 0.485i)T \)
41 \( 1 + (0.0275 - 0.999i)T \)
43 \( 1 + (0.938 + 0.345i)T \)
47 \( 1 + (0.635 + 0.771i)T \)
53 \( 1 + (-0.999 + 0.0110i)T \)
59 \( 1 + (-0.401 + 0.915i)T \)
61 \( 1 + (-0.917 + 0.396i)T \)
67 \( 1 + (0.509 + 0.860i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (-0.857 + 0.514i)T \)
79 \( 1 + (-0.480 + 0.876i)T \)
83 \( 1 + (-0.968 - 0.250i)T \)
89 \( 1 + (0.266 + 0.963i)T \)
97 \( 1 + (0.287 + 0.957i)T \)
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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

−23.275973989729476788036129465652, −22.5321021493947804767580306392, −21.69134589325741878675244633373, −20.14527403304809730488106521859, −19.32339258009810793454910199386, −18.83619700451385520978839148951, −18.33773495084315037323497282075, −17.06516632537892502575667639564, −16.44754541227405138623946798091, −15.24569451745084357337126812007, −14.64938745990633505135589004122, −14.20557295549121226790055595945, −12.71857385977430798766265611297, −12.26287056120321290383097124042, −11.17511701008571522570264854213, −9.696244783481579483961724754784, −8.901835576995855895331586598756, −8.07940283712617398418855311144, −7.44067927730051635341239839556, −6.44837011283635360132234893230, −5.80289448753984819577741162886, −4.32823290613672442035865813083, −3.25043694942764454004243082011, −1.94691604597514348105750341274, −0.32142074439590164445682181350, 1.33022922828743824193493608480, 2.800690240455321527905739479740, 3.85741117520919579497872378570, 4.19407458538492278212223436109, 5.23231825268702884239947317591, 7.312707301284458904306301073896, 7.84166572109449723606653124006, 9.0598580920696979402997031710, 9.597326645250974376261180967823, 10.55417249370213403191716392964, 11.26385558507245557037714452221, 12.24411438344051770708689793969, 13.02842550649372641695109544008, 14.247725809879100778392104252727, 14.78526343703087059301580254100, 16.01448630288376070827332115856, 16.950562956115115755490806350438, 17.250828987293605647520845922966, 18.905156544066016049268504286, 19.641877011429543732945991837943, 20.001050572645212070375595235143, 20.67953895873144394619207465234, 21.67850210597610793900014243271, 22.4116486857567722010318414552, 23.16379153135579031243891719772

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