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Properties

Label 1-571-571.168-r1-0-0
Degree $1$
Conductor $571$
Sign $0.956 + 0.292i$
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  + (0.754 + 0.656i)2-s + (−0.159 + 0.987i)3-s + (0.137 + 0.990i)4-s + (0.0935 + 0.995i)5-s + (−0.768 + 0.639i)6-s + (−0.701 − 0.712i)7-s + (−0.546 + 0.837i)8-s + (−0.949 − 0.314i)9-s + (−0.583 + 0.812i)10-s + (0.411 − 0.911i)11-s + (−0.999 − 0.0220i)12-s + (0.959 − 0.282i)13-s + (−0.0605 − 0.998i)14-s + (−0.997 − 0.0660i)15-s + (−0.962 + 0.272i)16-s + (0.0385 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.754 + 0.656i)2-s + (−0.159 + 0.987i)3-s + (0.137 + 0.990i)4-s + (0.0935 + 0.995i)5-s + (−0.768 + 0.639i)6-s + (−0.701 − 0.712i)7-s + (−0.546 + 0.837i)8-s + (−0.949 − 0.314i)9-s + (−0.583 + 0.812i)10-s + (0.411 − 0.911i)11-s + (−0.999 − 0.0220i)12-s + (0.959 − 0.282i)13-s + (−0.0605 − 0.998i)14-s + (−0.997 − 0.0660i)15-s + (−0.962 + 0.272i)16-s + (0.0385 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.956 + 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.818451241 + 0.2716924024i\)
\(L(\frac12)\) \(\approx\) \(1.818451241 + 0.2716924024i\)
\(L(1)\) \(\approx\) \(1.096276319 + 0.7315596672i\)
\(L(1)\) \(\approx\) \(1.096276319 + 0.7315596672i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.754 + 0.656i)T \)
3 \( 1 + (-0.159 + 0.987i)T \)
5 \( 1 + (0.0935 + 0.995i)T \)
7 \( 1 + (-0.701 - 0.712i)T \)
11 \( 1 + (0.411 - 0.911i)T \)
13 \( 1 + (0.959 - 0.282i)T \)
17 \( 1 + (0.0385 - 0.999i)T \)
19 \( 1 + (0.709 - 0.705i)T \)
23 \( 1 + (0.0495 - 0.998i)T \)
29 \( 1 + (0.851 + 0.523i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (-0.942 + 0.335i)T \)
41 \( 1 + (-0.975 - 0.218i)T \)
43 \( 1 + (0.952 + 0.303i)T \)
47 \( 1 + (-0.716 - 0.697i)T \)
53 \( 1 + (-0.391 - 0.920i)T \)
59 \( 1 + (-0.986 - 0.164i)T \)
61 \( 1 + (0.731 - 0.681i)T \)
67 \( 1 + (0.992 + 0.120i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.761 - 0.648i)T \)
79 \( 1 + (-0.970 - 0.240i)T \)
83 \( 1 + (-0.170 - 0.985i)T \)
89 \( 1 + (-0.938 - 0.345i)T \)
97 \( 1 + (-0.899 + 0.436i)T \)
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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

−23.16014414537067508871759192174, −22.25376373395502855690420269439, −21.30916735057672530897543413985, −20.49793600911581748666135469493, −19.611057371777169532366643120611, −19.16680857045810263892140761898, −18.13403812313137885219982659462, −17.31149932323794130113265731866, −16.110287445237464684009795248791, −15.39292764550742078884290569230, −14.12599286756710624382466098609, −13.466716238796054567987637686787, −12.51141644054503356785636419643, −12.29503892593096549324958903818, −11.43513278946269587599881290347, −10.11459464744381945758668639210, −9.18836642032167152258296093178, −8.32451383032406337210425813703, −6.91247007484786471524491122949, −5.99973733445859927759684926107, −5.4195634179748770579175233135, −4.161283816793936882731295859043, −3.073385361641949799732812388115, −1.7276031970023346666496879592, −1.27481454410718760284137176050, 0.35783949294434598841666636969, 2.94676567482802266977979552588, 3.30330402584531529427212768300, 4.26008275117662976769194737998, 5.39878352282458190786577043181, 6.36365282908964277051812015887, 6.89615194878816117938833711692, 8.209932256766388213701867172342, 9.21031458741976841389921158703, 10.29067907878554214043469072344, 11.12928645042938519083780454597, 11.784122760414361691702733156177, 13.2958337472194649256070728605, 13.941086349518553720415941236691, 14.5368477136318940336240366509, 15.76393638872562616826985235015, 15.983282646119049423452024333749, 16.93291045374237007906058734615, 17.78901009579914722879588165810, 18.83290060991408224187937530579, 20.104905917621800481398876297300, 20.77034131461042550003885488433, 21.763289024223780656252103963517, 22.42498918532960023985747562220, 22.81698176583585380404891197221

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