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Properties

Label 1-571-571.41-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.933 - 0.358i$
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  + (0.998 − 0.0550i)2-s + (−0.635 − 0.771i)3-s + (0.993 − 0.110i)4-s + (0.137 + 0.990i)5-s + (−0.677 − 0.735i)6-s + (0.879 − 0.475i)7-s + (0.986 − 0.164i)8-s + (−0.191 + 0.981i)9-s + (0.191 + 0.981i)10-s + (−0.754 − 0.656i)11-s + (−0.716 − 0.697i)12-s + (−0.821 + 0.569i)13-s + (0.851 − 0.523i)14-s + (0.677 − 0.735i)15-s + (0.975 − 0.218i)16-s + (−0.975 + 0.218i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0550i)2-s + (−0.635 − 0.771i)3-s + (0.993 − 0.110i)4-s + (0.137 + 0.990i)5-s + (−0.677 − 0.735i)6-s + (0.879 − 0.475i)7-s + (0.986 − 0.164i)8-s + (−0.191 + 0.981i)9-s + (0.191 + 0.981i)10-s + (−0.754 − 0.656i)11-s + (−0.716 − 0.697i)12-s + (−0.821 + 0.569i)13-s + (0.851 − 0.523i)14-s + (0.677 − 0.735i)15-s + (0.975 − 0.218i)16-s + (−0.975 + 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.933 - 0.358i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2472838012 - 1.335000857i\)
\(L(\frac12)\) \(\approx\) \(0.2472838012 - 1.335000857i\)
\(L(1)\) \(\approx\) \(1.320955458 - 0.3885521416i\)
\(L(1)\) \(\approx\) \(1.320955458 - 0.3885521416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.998 - 0.0550i)T \)
3 \( 1 + (-0.635 - 0.771i)T \)
5 \( 1 + (0.137 + 0.990i)T \)
7 \( 1 + (0.879 - 0.475i)T \)
11 \( 1 + (-0.754 - 0.656i)T \)
13 \( 1 + (-0.821 + 0.569i)T \)
17 \( 1 + (-0.975 + 0.218i)T \)
19 \( 1 + (-0.635 - 0.771i)T \)
23 \( 1 + (-0.986 - 0.164i)T \)
29 \( 1 + (0.904 + 0.426i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.821 - 0.569i)T \)
41 \( 1 + (-0.137 - 0.990i)T \)
43 \( 1 + (-0.191 - 0.981i)T \)
47 \( 1 + (0.298 - 0.954i)T \)
53 \( 1 + (0.998 + 0.0550i)T \)
59 \( 1 + (-0.879 + 0.475i)T \)
61 \( 1 + (0.451 - 0.892i)T \)
67 \( 1 + (-0.451 - 0.892i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.904 + 0.426i)T \)
79 \( 1 + (0.592 - 0.805i)T \)
83 \( 1 + (-0.298 + 0.954i)T \)
89 \( 1 + (-0.975 + 0.218i)T \)
97 \( 1 + (0.993 - 0.110i)T \)
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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

−23.41836634818408651722629191656, −22.51462385144951705235190941352, −21.67632774612160327312231843101, −21.11643441825042796690620647906, −20.44099450487406587340444849426, −19.76059522554513122899594282627, −17.96907611499251257040261307886, −17.44384887980113821968370219635, −16.47797255716979625725006892913, −15.685425825425762767697604884, −15.111718954983776260751877536207, −14.23236613216204170011740292504, −13.01279948552097857342370727632, −12.27214151445902387310551326528, −11.73003809009754676236638527290, −10.61455408097062488014981671609, −9.88678302947890703289586080295, −8.56806088511064293954148277473, −7.67370816167987106599743477373, −6.27124042876305406398365765483, −5.37613263421028206645973968573, −4.78513539831478793897267125932, −4.21830391773081579038715176320, −2.66443980150712256258242267486, −1.567816416528371449991278892009, 0.21887710492714305768442499527, 1.98449916297431306831606575443, 2.418946329570822225873292036853, 3.94681170089734515297897750, 4.968012558178264771661696204414, 5.83535576710858641401945926100, 6.86333692435925806576939418428, 7.28719813204977731515434102277, 8.380451705599701845001872094413, 10.40566364572616797981025545377, 10.8285444402191331093933979658, 11.56527552644047899886847368633, 12.405101689115587498315824332593, 13.60820724650717407667300867636, 13.865641808986011203729452287002, 14.862209944729895404641365432775, 15.76958362451589451367911795634, 16.88744598338195997159226610732, 17.61954128976339436134601850222, 18.51058024809404424185533539520, 19.38487684639565662873781720591, 20.16092305012856702593930645075, 21.54517964578879823765208025010, 21.80433866675814455095233028308, 22.70273359004618142848724817029

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