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Properties

Label 1-571-571.175-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.730 + 0.683i$
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  + (0.592 − 0.805i)2-s + (0.754 − 0.656i)3-s + (−0.298 − 0.954i)4-s + (0.716 − 0.697i)5-s + (−0.0825 − 0.996i)6-s + (−0.546 − 0.837i)7-s + (−0.945 − 0.324i)8-s + (0.137 − 0.990i)9-s + (−0.137 − 0.990i)10-s + (−0.926 + 0.376i)11-s + (−0.851 − 0.523i)12-s + (0.635 − 0.771i)13-s + (−0.998 − 0.0550i)14-s + (0.0825 − 0.996i)15-s + (−0.821 + 0.569i)16-s + (0.821 − 0.569i)17-s + ⋯
L(s)  = 1  + (0.592 − 0.805i)2-s + (0.754 − 0.656i)3-s + (−0.298 − 0.954i)4-s + (0.716 − 0.697i)5-s + (−0.0825 − 0.996i)6-s + (−0.546 − 0.837i)7-s + (−0.945 − 0.324i)8-s + (0.137 − 0.990i)9-s + (−0.137 − 0.990i)10-s + (−0.926 + 0.376i)11-s + (−0.851 − 0.523i)12-s + (0.635 − 0.771i)13-s + (−0.998 − 0.0550i)14-s + (0.0825 − 0.996i)15-s + (−0.821 + 0.569i)16-s + (0.821 − 0.569i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.730 + 0.683i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.730 + 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.371970193 - 3.473071337i\)
\(L(\frac12)\) \(\approx\) \(-1.371970193 - 3.473071337i\)
\(L(1)\) \(\approx\) \(0.8544334603 - 1.702287607i\)
\(L(1)\) \(\approx\) \(0.8544334603 - 1.702287607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.592 - 0.805i)T \)
3 \( 1 + (0.754 - 0.656i)T \)
5 \( 1 + (0.716 - 0.697i)T \)
7 \( 1 + (-0.546 - 0.837i)T \)
11 \( 1 + (-0.926 + 0.376i)T \)
13 \( 1 + (0.635 - 0.771i)T \)
17 \( 1 + (0.821 - 0.569i)T \)
19 \( 1 + (0.754 - 0.656i)T \)
23 \( 1 + (0.945 - 0.324i)T \)
29 \( 1 + (0.350 + 0.936i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (0.635 + 0.771i)T \)
41 \( 1 + (-0.716 + 0.697i)T \)
43 \( 1 + (0.137 + 0.990i)T \)
47 \( 1 + (-0.904 - 0.426i)T \)
53 \( 1 + (0.592 + 0.805i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (0.993 + 0.110i)T \)
67 \( 1 + (-0.993 + 0.110i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.350 + 0.936i)T \)
79 \( 1 + (-0.975 + 0.218i)T \)
83 \( 1 + (0.904 + 0.426i)T \)
89 \( 1 + (0.821 - 0.569i)T \)
97 \( 1 + (-0.298 - 0.954i)T \)
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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

−23.43691440635682171960548564495, −22.67594799424995415769540806814, −21.77761168182409470172022734079, −21.28444180113250316801172362065, −20.780491236666921006000299563802, −19.07620738176673653236865028171, −18.681588735445760758489582595553, −17.60550837091739342668165845666, −16.39553068731358980902890789599, −15.93071965501374345651296565752, −15.042530682204829538955738053296, −14.36590954753804149975323752540, −13.59455564299947498354598688845, −12.93977841878077526216260044213, −11.68229941074690864411156738817, −10.48124382854395050007158090225, −9.5669648875379870164371805361, −8.76439089644391953225871916282, −7.88581084007038548499327649384, −6.795088309667869100934402176127, −5.74779082178809082641054910500, −5.20028427546510466394010119245, −3.659659769550741734284931222578, −3.106498632728454806064613332904, −2.10911241022732429536417177434, 0.693978111350181176090724924171, 1.26892267962908526185468392837, 2.69047265578425565636720596463, 3.23021862491132747058345283547, 4.58474309868443210270777590407, 5.513601479860817850330451653371, 6.57797561589900779095196143939, 7.65960684117826417312683951685, 8.79442372045219741443411161396, 9.74248970411034923323053924937, 10.26779471277732032857708747897, 11.55149771732629053841187517890, 12.74879090181392500667898145172, 13.14015719769084597924363026060, 13.612418060534705982508519861617, 14.57291817413165569269106336347, 15.56309163758641719109556102299, 16.63352415905589743453311072139, 17.97671458002035053757043276786, 18.34659835594717062707132847460, 19.489378396384085797822474972509, 20.37794737908091499883810279639, 20.532544768731449451201120865396, 21.398912482478004569994801056054, 22.63314193577809748273775360730

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