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Properties

Label 1-571-571.153-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.821 + 0.570i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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 + (−0.137 − 0.990i)3-s + (−0.821 − 0.569i)4-s + (0.0275 + 0.999i)5-s + (−0.986 − 0.164i)6-s + (0.401 + 0.915i)7-s + (−0.789 + 0.614i)8-s + (−0.962 + 0.272i)9-s + (0.962 + 0.272i)10-s + (0.716 + 0.697i)11-s + (−0.451 + 0.892i)12-s + (−0.191 + 0.981i)13-s + (0.993 − 0.110i)14-s + (0.986 − 0.164i)15-s + (0.350 + 0.936i)16-s + (−0.350 − 0.936i)17-s + ⋯
L(s)  = 1  + (0.298 − 0.954i)2-s + (−0.137 − 0.990i)3-s + (−0.821 − 0.569i)4-s + (0.0275 + 0.999i)5-s + (−0.986 − 0.164i)6-s + (0.401 + 0.915i)7-s + (−0.789 + 0.614i)8-s + (−0.962 + 0.272i)9-s + (0.962 + 0.272i)10-s + (0.716 + 0.697i)11-s + (−0.451 + 0.892i)12-s + (−0.191 + 0.981i)13-s + (0.993 − 0.110i)14-s + (0.986 − 0.164i)15-s + (0.350 + 0.936i)16-s + (−0.350 − 0.936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.821 + 0.570i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1414559078 - 0.4515059713i\)
\(L(\frac12)\) \(\approx\) \(-0.1414559078 - 0.4515059713i\)
\(L(1)\) \(\approx\) \(0.7777471390 - 0.4740938250i\)
\(L(1)\) \(\approx\) \(0.7777471390 - 0.4740938250i\)

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 \( 1 + (-0.137 - 0.990i)T \)
5 \( 1 + (0.0275 + 0.999i)T \)
7 \( 1 + (0.401 + 0.915i)T \)
11 \( 1 + (0.716 + 0.697i)T \)
13 \( 1 + (-0.191 + 0.981i)T \)
17 \( 1 + (-0.350 - 0.936i)T \)
19 \( 1 + (-0.137 - 0.990i)T \)
23 \( 1 + (0.789 + 0.614i)T \)
29 \( 1 + (-0.754 - 0.656i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.191 - 0.981i)T \)
41 \( 1 + (-0.0275 - 0.999i)T \)
43 \( 1 + (-0.962 - 0.272i)T \)
47 \( 1 + (-0.635 + 0.771i)T \)
53 \( 1 + (0.298 + 0.954i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.975 - 0.218i)T \)
67 \( 1 + (-0.975 - 0.218i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.754 - 0.656i)T \)
79 \( 1 + (-0.904 - 0.426i)T \)
83 \( 1 + (0.635 - 0.771i)T \)
89 \( 1 + (-0.350 - 0.936i)T \)
97 \( 1 + (-0.821 - 0.569i)T \)
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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

−23.558669336748721919136657315894, −22.79821614267539798475090699036, −21.952896747136690771360154423579, −21.182578814223817526969360533061, −20.40257295768869911467869450020, −19.64747448796717469062696645502, −18.124928605743147039167135356910, −17.08842634370417760117866746470, −16.78811710854645813327228786458, −16.21691649320562795227107269660, −14.96259902373858946161310112788, −14.60892351734536509102125485780, −13.448733176440694999285457759983, −12.73745903461363467762821984544, −11.58480277952197329183095222590, −10.49749391505964724542800401152, −9.59205883930096601198913602730, −8.49644934994965889936175929571, −8.18302516625474534988816669807, −6.72418041648821179220382917105, −5.64829634953129234387802950358, −4.9880964537467872715881768424, −4.03330769043559492824262194530, −3.4431038580753046269623979054, −1.134124762977427006064908985667, 0.11355766616581244116662679602, 1.79760592431820798134101737048, 2.23143366949663681984076807513, 3.285741812449989992285685417239, 4.655675743930300647695248655994, 5.67419977701404947360083467051, 6.70397235110975123965695708605, 7.46519719440390107989568270798, 8.99481923833989769465850164096, 9.41649064577908983736368567901, 10.999298476013336766406264067661, 11.47746347372526805941990195009, 12.07802882544830042151063045332, 13.08751758289815880616673986426, 13.98131229300947658366388417842, 14.61337424183376599430200532646, 15.3972748827717406925705513561, 17.1726173406310310486511157863, 17.87429996612610586787184181726, 18.52641342438973899831331332431, 19.17222884216837755485404529796, 19.82122196842566516038550283280, 20.93077189644347463779154787318, 21.93068692036222642617312426722, 22.39752322966466185927678164704

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