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Properties

Label 1-571-571.173-r0-0-0
Degree $1$
Conductor $571$
Sign $0.932 - 0.360i$
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 + (0.984 + 0.175i)3-s + (−0.821 − 0.569i)4-s + (0.565 − 0.824i)5-s + (−0.461 + 0.887i)6-s + (−0.995 + 0.0990i)7-s + (0.789 − 0.614i)8-s + (0.938 + 0.345i)9-s + (0.618 + 0.785i)10-s + (−0.989 − 0.142i)11-s + (−0.709 − 0.705i)12-s + (0.731 − 0.681i)13-s + (0.202 − 0.979i)14-s + (0.701 − 0.712i)15-s + (0.350 + 0.936i)16-s + (−0.834 − 0.551i)17-s + ⋯
L(s)  = 1  + (−0.298 + 0.954i)2-s + (0.984 + 0.175i)3-s + (−0.821 − 0.569i)4-s + (0.565 − 0.824i)5-s + (−0.461 + 0.887i)6-s + (−0.995 + 0.0990i)7-s + (0.789 − 0.614i)8-s + (0.938 + 0.345i)9-s + (0.618 + 0.785i)10-s + (−0.989 − 0.142i)11-s + (−0.709 − 0.705i)12-s + (0.731 − 0.681i)13-s + (0.202 − 0.979i)14-s + (0.701 − 0.712i)15-s + (0.350 + 0.936i)16-s + (−0.834 − 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.318970792 - 0.2457551806i\)
\(L(\frac12)\) \(\approx\) \(1.318970792 - 0.2457551806i\)
\(L(1)\) \(\approx\) \(1.101510175 + 0.1546872051i\)
\(L(1)\) \(\approx\) \(1.101510175 + 0.1546872051i\)

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 \( 1 + (0.984 + 0.175i)T \)
5 \( 1 + (0.565 - 0.824i)T \)
7 \( 1 + (-0.995 + 0.0990i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
13 \( 1 + (0.731 - 0.681i)T \)
17 \( 1 + (-0.834 - 0.551i)T \)
19 \( 1 + (0.471 - 0.882i)T \)
23 \( 1 + (0.828 - 0.560i)T \)
29 \( 1 + (-0.754 - 0.656i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.421 + 0.906i)T \)
41 \( 1 + (0.0275 + 0.999i)T \)
43 \( 1 + (-0.0385 - 0.999i)T \)
47 \( 1 + (0.635 - 0.771i)T \)
53 \( 1 + (0.802 + 0.596i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.0935 - 0.995i)T \)
67 \( 1 + (-0.917 + 0.396i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.996 - 0.0880i)T \)
79 \( 1 + (0.685 - 0.728i)T \)
83 \( 1 + (-0.537 - 0.843i)T \)
89 \( 1 + (0.999 - 0.0440i)T \)
97 \( 1 + (0.329 + 0.944i)T \)
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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

−23.03071116292169561363363839576, −22.30999744094594067073261317174, −21.29528178925173221638834481502, −20.915635167657984288817628187708, −19.829389798365863356572979854838, −19.1951797558349902812375282719, −18.43338820282121986046229984847, −18.007156388676593947802730641865, −16.659046590733815985285733358413, −15.658511751028075200800388781644, −14.59433586153325049871098501841, −13.66334984739264049595483156858, −13.19106588511825455152165544017, −12.43253891426715014713361682202, −10.949334498665798119293149585166, −10.420416069609930478351623411859, −9.40895231835094911715808464390, −8.95333868820368898386322153424, −7.65643176091559312627520300191, −6.90619156015612816733018769687, −5.599105892226961881367766896200, −3.93977571808958229869484850005, −3.322669360176634626527012358038, −2.41252359904509583477458132282, −1.56894364841283287781462939435, 0.68406225822172541739504883269, 2.281874073653095954775218089300, 3.451753562752688361504909972386, 4.72083568289806564515012954792, 5.50025674968354195826549655198, 6.61149637833963395923080639940, 7.575089060590428835336857548355, 8.59007647983452573433477602121, 9.10765555216448189905342998898, 9.856449863447544893349689864492, 10.74412986675855158822125927918, 12.731926985255160966452644570344, 13.450881773560881715437088391011, 13.53904168385463631951359604504, 15.07400186042168678916841204947, 15.68059173981875101837016636412, 16.22979216240196749127503329714, 17.18504200732969838196884995109, 18.32359354626314820879455841512, 18.76988499889833730125228806974, 20.04857053648596306322154839494, 20.40371717463723002086811612301, 21.60826841142298780159583971155, 22.393603201590045300023487358476, 23.43455264316803266518353280152

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