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Properties

Label 1-571-571.17-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: $\chi_{571} (17, \cdot )$
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

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

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