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Properties

Label 1-571-571.258-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.862 - 0.506i$
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  + (0.635 + 0.771i)2-s + (0.0275 − 0.999i)3-s + (−0.191 + 0.981i)4-s + (−0.592 + 0.805i)5-s + (0.789 − 0.614i)6-s + (−0.0825 + 0.996i)7-s + (−0.879 + 0.475i)8-s + (−0.998 − 0.0550i)9-s + (−0.998 + 0.0550i)10-s + (0.451 + 0.892i)11-s + (0.975 + 0.218i)12-s + (−0.962 − 0.272i)13-s + (−0.821 + 0.569i)14-s + (0.789 + 0.614i)15-s + (−0.926 − 0.376i)16-s + (−0.926 − 0.376i)17-s + ⋯
L(s)  = 1  + (0.635 + 0.771i)2-s + (0.0275 − 0.999i)3-s + (−0.191 + 0.981i)4-s + (−0.592 + 0.805i)5-s + (0.789 − 0.614i)6-s + (−0.0825 + 0.996i)7-s + (−0.879 + 0.475i)8-s + (−0.998 − 0.0550i)9-s + (−0.998 + 0.0550i)10-s + (0.451 + 0.892i)11-s + (0.975 + 0.218i)12-s + (−0.962 − 0.272i)13-s + (−0.821 + 0.569i)14-s + (0.789 + 0.614i)15-s + (−0.926 − 0.376i)16-s + (−0.926 − 0.376i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.862 - 0.506i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.862 - 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1216055921 + 0.4470749584i\)
\(L(\frac12)\) \(\approx\) \(-0.1216055921 + 0.4470749584i\)
\(L(1)\) \(\approx\) \(0.7863800575 + 0.4434113204i\)
\(L(1)\) \(\approx\) \(0.7863800575 + 0.4434113204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.635 + 0.771i)T \)
3 \( 1 + (0.0275 - 0.999i)T \)
5 \( 1 + (-0.592 + 0.805i)T \)
7 \( 1 + (-0.0825 + 0.996i)T \)
11 \( 1 + (0.451 + 0.892i)T \)
13 \( 1 + (-0.962 - 0.272i)T \)
17 \( 1 + (-0.926 - 0.376i)T \)
19 \( 1 + (0.0275 - 0.999i)T \)
23 \( 1 + (-0.879 - 0.475i)T \)
29 \( 1 + (0.716 - 0.697i)T \)
31 \( 1 + (0.789 - 0.614i)T \)
37 \( 1 + (-0.962 + 0.272i)T \)
41 \( 1 + (-0.592 + 0.805i)T \)
43 \( 1 + (-0.998 + 0.0550i)T \)
47 \( 1 + (0.137 + 0.990i)T \)
53 \( 1 + (0.635 - 0.771i)T \)
59 \( 1 + (-0.0825 + 0.996i)T \)
61 \( 1 + (0.350 - 0.936i)T \)
67 \( 1 + (0.350 + 0.936i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.716 + 0.697i)T \)
79 \( 1 + (-0.754 + 0.656i)T \)
83 \( 1 + (0.137 + 0.990i)T \)
89 \( 1 + (-0.926 - 0.376i)T \)
97 \( 1 + (-0.191 + 0.981i)T \)
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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

−22.61706837309476894057120682323, −21.80702183338366820513515714835, −21.16779979002664350503833370141, −20.18707500829777607420413296659, −19.82918636184266435545512899431, −19.15060227882036121165744269532, −17.54740020558793051565820170673, −16.69526317025680139026115118796, −15.99242370868968292222510623334, −15.100856581998999977797972491518, −14.09121941165147790241366866685, −13.59892121763456861550743229851, −12.25315647938718035560959635445, −11.72432944738875402517423941476, −10.678797453054052140522547825759, −10.087849877375112589359434962338, −9.05950247281064967337560484700, −8.26247760972057901523917786307, −6.72764274899891881773459036195, −5.50191254787364401907894845713, −4.63530552761272963022423214644, −3.92672906356408611820329664117, −3.29145291358848184474663136518, −1.66599003861631871523469701288, −0.18001738271156744892441103362, 2.38351100268778359132410615766, 2.751539821040869702832584647774, 4.26464604294660730885282312356, 5.262709651900241270531427065478, 6.60647632970610250759360975926, 6.755177147947911948706395531446, 7.86213621311222475109611126817, 8.568120153799356094523237044467, 9.76925142231412155844541930854, 11.518952962233555062056834906355, 11.90338519270581457824269711507, 12.681396142996535303237524930491, 13.646487869036119562576097639844, 14.56322071021899818560042553439, 15.180377661488076515948322165661, 15.82736505462418526774377868968, 17.26631720275502692839656231136, 17.796351853677632187748505410486, 18.54121205421820354135835048181, 19.53879642910291853085993147844, 20.233072083972308720398658071346, 21.69878360658083024255251134840, 22.500484842695392207706681033721, 22.73050313398868954465381625344, 23.8817368097550214385271782385

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