↑

Properties

Label 1-571-571.112-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.280 + 0.959i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0275 + 0.999i)2-s + (0.126 + 0.991i)3-s + (−0.998 − 0.0550i)4-s + (0.996 − 0.0880i)5-s + (−0.995 + 0.0990i)6-s + (0.846 + 0.533i)7-s + (0.0825 − 0.996i)8-s + (−0.968 + 0.250i)9-s + (0.0605 + 0.998i)10-s + (−0.834 − 0.551i)11-s + (−0.0715 − 0.997i)12-s + (0.802 − 0.596i)13-s + (−0.556 + 0.831i)14-s + (0.213 + 0.976i)15-s + (0.993 + 0.110i)16-s + (0.868 − 0.495i)17-s + ⋯
L(s)  = 1  + (−0.0275 + 0.999i)2-s + (0.126 + 0.991i)3-s + (−0.998 − 0.0550i)4-s + (0.996 − 0.0880i)5-s + (−0.995 + 0.0990i)6-s + (0.846 + 0.533i)7-s + (0.0825 − 0.996i)8-s + (−0.968 + 0.250i)9-s + (0.0605 + 0.998i)10-s + (−0.834 − 0.551i)11-s + (−0.0715 − 0.997i)12-s + (0.802 − 0.596i)13-s + (−0.556 + 0.831i)14-s + (0.213 + 0.976i)15-s + (0.993 + 0.110i)16-s + (0.868 − 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.280 + 0.959i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.280 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.606830878 + 2.142691260i\)
\(L(\frac12)\) \(\approx\) \(1.606830878 + 2.142691260i\)
\(L(1)\) \(\approx\) \(1.031181958 + 0.9006627352i\)
\(L(1)\) \(\approx\) \(1.031181958 + 0.9006627352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.0275 + 0.999i)T \)
3 \( 1 + (0.126 + 0.991i)T \)
5 \( 1 + (0.996 - 0.0880i)T \)
7 \( 1 + (0.846 + 0.533i)T \)
11 \( 1 + (-0.834 - 0.551i)T \)
13 \( 1 + (0.802 - 0.596i)T \)
17 \( 1 + (0.868 - 0.495i)T \)
19 \( 1 + (0.982 + 0.186i)T \)
23 \( 1 + (-0.973 - 0.229i)T \)
29 \( 1 + (0.975 - 0.218i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (-0.319 + 0.947i)T \)
41 \( 1 + (0.754 - 0.656i)T \)
43 \( 1 + (-0.537 + 0.843i)T \)
47 \( 1 + (0.592 - 0.805i)T \)
53 \( 1 + (0.609 + 0.792i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (0.761 - 0.648i)T \)
67 \( 1 + (0.381 - 0.924i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.660 + 0.750i)T \)
79 \( 1 + (0.709 + 0.705i)T \)
83 \( 1 + (0.583 + 0.812i)T \)
89 \( 1 + (-0.411 + 0.911i)T \)
97 \( 1 + (0.775 + 0.631i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.896897129599647010980476248328, −21.68775204585971283528288676169, −20.99857045549373483351694010923, −20.39635246395596566138101903734, −19.52080249828363554346564888119, −18.452722582065696207427893520306, −17.938307066135961531839387525838, −17.54105266736698103601177405917, −16.35350432674623693401408502405, −14.57322297266546648631292452087, −14.010551772499307441590135579898, −13.45502271299904104497702871867, −12.55822968462321431052080387824, −11.74885700609539150776023000584, −10.75750607183200469676206370536, −10.05460226851660529189389194918, −8.93322912367533465211421057244, −8.07581212937701711022364132783, −7.16602490963720748436671405674, −5.811867909015706383619692908013, −5.056078136442888005041071545491, −3.67561639683130489826203264793, −2.48700956149177751720923642380, −1.658538056691510598066752582926, −0.96802662393707995955743416320, 0.86412557162904352810176658314, 2.57725790218370266888674237187, 3.72711608132841787184800713131, 5.139303008306559792211634363362, 5.3919952229016572552701648139, 6.22926345102512817660352469561, 7.94990754523634361014181916710, 8.34529338020957451991110796947, 9.43092709114611751789196519094, 10.07765109447928331287781882751, 11.005396051742708187516322783107, 12.31071964760420899344839912297, 13.73156530373844642203245836348, 13.94311015752784082100375756515, 14.970410965262307391723278117881, 15.775695057011834901802639681600, 16.39067051142152682758833224324, 17.30905473544824659002796652245, 18.170298363523338604999444475734, 18.60084288248875433539697264888, 20.31766879427808780981956272718, 21.017222540752411975184503106077, 21.662289528442213247618315931872, 22.39733701423431509469801527689, 23.28953530559781421787649408398

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