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Properties

Label 1-571-571.118-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.687 + 0.726i$
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  + (−0.975 + 0.218i)2-s + (0.926 + 0.376i)3-s + (0.904 − 0.426i)4-s + (0.851 − 0.523i)5-s + (−0.986 − 0.164i)6-s + (0.401 + 0.915i)7-s + (−0.789 + 0.614i)8-s + (0.716 + 0.697i)9-s + (−0.716 + 0.697i)10-s + (−0.962 + 0.272i)11-s + (0.998 − 0.0550i)12-s + (−0.754 − 0.656i)13-s + (−0.592 − 0.805i)14-s + (0.986 − 0.164i)15-s + (0.635 − 0.771i)16-s + (−0.635 + 0.771i)17-s + ⋯
L(s)  = 1  + (−0.975 + 0.218i)2-s + (0.926 + 0.376i)3-s + (0.904 − 0.426i)4-s + (0.851 − 0.523i)5-s + (−0.986 − 0.164i)6-s + (0.401 + 0.915i)7-s + (−0.789 + 0.614i)8-s + (0.716 + 0.697i)9-s + (−0.716 + 0.697i)10-s + (−0.962 + 0.272i)11-s + (0.998 − 0.0550i)12-s + (−0.754 − 0.656i)13-s + (−0.592 − 0.805i)14-s + (0.986 − 0.164i)15-s + (0.635 − 0.771i)16-s + (−0.635 + 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.687 + 0.726i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.687 + 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6364802920 + 1.477760673i\)
\(L(\frac12)\) \(\approx\) \(0.6364802920 + 1.477760673i\)
\(L(1)\) \(\approx\) \(0.9528028078 + 0.4012549483i\)
\(L(1)\) \(\approx\) \(0.9528028078 + 0.4012549483i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.975 + 0.218i)T \)
3 \( 1 + (0.926 + 0.376i)T \)
5 \( 1 + (0.851 - 0.523i)T \)
7 \( 1 + (0.401 + 0.915i)T \)
11 \( 1 + (-0.962 + 0.272i)T \)
13 \( 1 + (-0.754 - 0.656i)T \)
17 \( 1 + (-0.635 + 0.771i)T \)
19 \( 1 + (0.926 + 0.376i)T \)
23 \( 1 + (0.789 + 0.614i)T \)
29 \( 1 + (-0.191 + 0.981i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.754 + 0.656i)T \)
41 \( 1 + (-0.851 + 0.523i)T \)
43 \( 1 + (0.716 - 0.697i)T \)
47 \( 1 + (-0.350 - 0.936i)T \)
53 \( 1 + (-0.975 - 0.218i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (-0.298 + 0.954i)T \)
67 \( 1 + (0.298 + 0.954i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.191 + 0.981i)T \)
79 \( 1 + (0.821 - 0.569i)T \)
83 \( 1 + (0.350 + 0.936i)T \)
89 \( 1 + (-0.635 + 0.771i)T \)
97 \( 1 + (0.904 - 0.426i)T \)
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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

−22.62967803294729720486763437707, −21.44439215993436362829562702950, −20.86603829397525231572062519071, −20.22424356088822736296280875142, −19.30816634375175952370299205724, −18.52781812231716296101079703310, −17.90888572830237104045142546975, −17.156006109808268381580876216545, −16.112427683526627901651982318, −15.121835639601290694864942875506, −14.1340497871573090820256984734, −13.53102909384651875518551191830, −12.56038565245248718663875418730, −11.24783704161615594299857517673, −10.545322864697940583587800552387, −9.5692331461867720532452800974, −9.064357300246075807992590574570, −7.73948609757814823049780749941, −7.28376449661271021729658753082, −6.46299208003256256874562057995, −4.86770300826756897363810667598, −3.33128736783377112080839370326, −2.50652587954349386023185100187, −1.70742968930701358999447422078, −0.427953948758629833023003337630, 1.49804855583159832960503883964, 2.23132355169156835198349545213, 3.13403107991178257940348186621, 5.1221102870698626885606752783, 5.437950843475042093135100057972, 6.99659014135445932402968548879, 7.993268314373245830025817366667, 8.64446493784234263480576663045, 9.43323248716453683036845802721, 10.09258975973334835717938757040, 10.9562035204189683899031856531, 12.360904410538761818929616703471, 13.12999891787570137319367267203, 14.32693776062409976160333801302, 15.14058150046643926650313933658, 15.67050377874510466816243540100, 16.6768832751757283755432474835, 17.63874359792894560172027499597, 18.28050390595459895305023108242, 19.09781909560362218369746289560, 20.205752729930894648737165779912, 20.56032637947513296342691621248, 21.49186858708833939033062679531, 22.09659065306753609269305905938, 23.88111608290382330245230014506

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