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Properties

Label 1-571-571.120-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.579 - 0.814i$
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  + (−0.0825 − 0.996i)2-s + (−0.768 − 0.639i)3-s + (−0.986 + 0.164i)4-s + (−0.627 − 0.778i)5-s + (−0.574 + 0.818i)6-s + (0.980 + 0.197i)7-s + (0.245 + 0.969i)8-s + (0.180 + 0.983i)9-s + (−0.724 + 0.689i)10-s + (−0.724 − 0.689i)11-s + (0.863 + 0.504i)12-s + (0.828 − 0.560i)13-s + (0.115 − 0.993i)14-s + (−0.0165 + 0.999i)15-s + (0.945 − 0.324i)16-s + (0.601 + 0.799i)17-s + ⋯
L(s)  = 1  + (−0.0825 − 0.996i)2-s + (−0.768 − 0.639i)3-s + (−0.986 + 0.164i)4-s + (−0.627 − 0.778i)5-s + (−0.574 + 0.818i)6-s + (0.980 + 0.197i)7-s + (0.245 + 0.969i)8-s + (0.180 + 0.983i)9-s + (−0.724 + 0.689i)10-s + (−0.724 − 0.689i)11-s + (0.863 + 0.504i)12-s + (0.828 − 0.560i)13-s + (0.115 − 0.993i)14-s + (−0.0165 + 0.999i)15-s + (0.945 − 0.324i)16-s + (0.601 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.579 - 0.814i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.579 - 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4329207216 - 0.8392232384i\)
\(L(\frac12)\) \(\approx\) \(0.4329207216 - 0.8392232384i\)
\(L(1)\) \(\approx\) \(0.5811888820 - 0.5474333836i\)
\(L(1)\) \(\approx\) \(0.5811888820 - 0.5474333836i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.0825 - 0.996i)T \)
3 \( 1 + (-0.768 - 0.639i)T \)
5 \( 1 + (-0.627 - 0.778i)T \)
7 \( 1 + (0.980 + 0.197i)T \)
11 \( 1 + (-0.724 - 0.689i)T \)
13 \( 1 + (0.828 - 0.560i)T \)
17 \( 1 + (0.601 + 0.799i)T \)
19 \( 1 + (0.997 + 0.0660i)T \)
23 \( 1 + (0.371 + 0.928i)T \)
29 \( 1 + (0.789 + 0.614i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (-0.340 - 0.940i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (0.431 + 0.901i)T \)
47 \( 1 + (0.945 - 0.324i)T \)
53 \( 1 + (-0.973 + 0.229i)T \)
59 \( 1 + (-0.677 - 0.735i)T \)
61 \( 1 + (0.652 + 0.757i)T \)
67 \( 1 + (-0.973 + 0.229i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.340 - 0.940i)T \)
79 \( 1 + (0.894 - 0.446i)T \)
83 \( 1 + (-0.574 + 0.818i)T \)
89 \( 1 + (-0.574 + 0.818i)T \)
97 \( 1 + (-0.148 + 0.988i)T \)
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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

−23.42266917494846811317431502891, −22.94084885632203202702067316883, −22.23716614244466025105731464667, −21.13456881086536016893441572503, −20.4920651388227080708357235954, −18.8988640709538741223343891442, −18.23867618466020795981986702461, −17.70741706401828631785579932505, −16.7158370470864316180039233991, −15.79274649190058696882016391210, −15.425665839909688994740549389453, −14.42093262735947671298622324583, −13.794706209632140809061514375, −12.279645552431647662293904289234, −11.48913740196916244724883271354, −10.5446925565493611773641220416, −9.84979126989418892786419824918, −8.61621309620053869228926082648, −7.65227815674643343014493730177, −6.92113042861328706814420065532, −5.9474005792883419925109047529, −4.807998741810519870884902530783, −4.38394780482358244348753342468, −3.08580866834320871381124890389, −0.95839745460266320398424462840, 0.87720355126805483394764208511, 1.48323294391840929446645191035, 2.97396918347193428404622601084, 4.17062456116827067413569256713, 5.27671678374713708688043040752, 5.676253655520766656604754343395, 7.699841733739660468026185049559, 8.06162787114517990832020421084, 8.99395724311758681404329990691, 10.48638573485582829543817750517, 11.07263618578636226732348597789, 11.836863975340070391025455009380, 12.49936109599716046381794401022, 13.32255735454010191325846367635, 14.07390859666753675105513340174, 15.54451941568364676264882354863, 16.402085552956306092310534047375, 17.48846845650210590170996760095, 17.9343492220010644318792299083, 18.91342349924348281604500959512, 19.46385993936531796068246066118, 20.628924261514438166551524968903, 21.12396281699648603003446713392, 22.01656881715618261240504191018, 23.156728883899820516914948158675

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