↑

Properties

Label 1-571-571.157-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.338 - 0.941i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.191 − 0.981i)2-s + (0.775 − 0.631i)3-s + (−0.926 + 0.376i)4-s + (0.815 + 0.578i)5-s + (−0.768 − 0.639i)6-s + (0.701 − 0.712i)7-s + (0.546 + 0.837i)8-s + (0.202 − 0.979i)9-s + (0.411 − 0.911i)10-s + (0.583 − 0.812i)11-s + (−0.480 + 0.876i)12-s + (−0.234 − 0.972i)13-s + (−0.834 − 0.551i)14-s + (0.997 − 0.0660i)15-s + (0.716 − 0.697i)16-s + (0.884 + 0.466i)17-s + ⋯
L(s)  = 1  + (−0.191 − 0.981i)2-s + (0.775 − 0.631i)3-s + (−0.926 + 0.376i)4-s + (0.815 + 0.578i)5-s + (−0.768 − 0.639i)6-s + (0.701 − 0.712i)7-s + (0.546 + 0.837i)8-s + (0.202 − 0.979i)9-s + (0.411 − 0.911i)10-s + (0.583 − 0.812i)11-s + (−0.480 + 0.876i)12-s + (−0.234 − 0.972i)13-s + (−0.834 − 0.551i)14-s + (0.997 − 0.0660i)15-s + (0.716 − 0.697i)16-s + (0.884 + 0.466i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.338 - 0.941i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.338 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.130499460 - 1.607873201i\)
\(L(\frac12)\) \(\approx\) \(1.130499460 - 1.607873201i\)
\(L(1)\) \(\approx\) \(1.132493264 - 0.8948492948i\)
\(L(1)\) \(\approx\) \(1.132493264 - 0.8948492948i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.191 - 0.981i)T \)
3 \( 1 + (0.775 - 0.631i)T \)
5 \( 1 + (0.815 + 0.578i)T \)
7 \( 1 + (0.701 - 0.712i)T \)
11 \( 1 + (0.583 - 0.812i)T \)
13 \( 1 + (-0.234 - 0.972i)T \)
17 \( 1 + (0.884 + 0.466i)T \)
19 \( 1 + (-0.256 + 0.966i)T \)
23 \( 1 + (0.0495 + 0.998i)T \)
29 \( 1 + (0.0275 + 0.999i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (0.761 - 0.648i)T \)
41 \( 1 + (-0.298 + 0.954i)T \)
43 \( 1 + (-0.739 - 0.673i)T \)
47 \( 1 + (-0.962 - 0.272i)T \)
53 \( 1 + (-0.992 + 0.120i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (0.224 - 0.974i)T \)
67 \( 1 + (0.391 - 0.920i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.942 - 0.335i)T \)
79 \( 1 + (-0.693 - 0.720i)T \)
83 \( 1 + (0.938 - 0.345i)T \)
89 \( 1 + (-0.170 + 0.985i)T \)
97 \( 1 + (0.0715 + 0.997i)T \)
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

−23.85818934998744221169644874740, −22.53942195641198812905914201829, −21.831819452977686403132189432417, −21.09731384454948233951719233343, −20.31705404907597287903237892184, −19.16861598171551009897282989418, −18.43017981153721588335918982252, −17.3603768445906346977196120393, −16.80661877558895091684555755717, −15.90531411472753250419760561487, −14.93413793345875240332780564534, −14.46148717628438724231519210533, −13.69290163730577324223702516246, −12.736060762348274298923482377398, −11.51728132464336382557107895811, −9.96762837260158993458811319424, −9.52095326972232546674258059603, −8.776904618579894834710470340808, −8.04323268789523634441014254922, −6.89713471630901364938914625054, −5.8005984900836008639141328509, −4.67794897929442013630692698272, −4.44298193029039022490565529481, −2.54842920181637665130993075003, −1.52591996279291923682702204766, 1.2232058983964804460667690421, 1.75350250450089261521026580904, 3.19883099385481239394737559448, 3.53328218198063245285302939527, 5.14855248117689615050656121701, 6.295666367345689930451458325422, 7.61427997322980720937828639148, 8.188137881511573085886455059530, 9.253956742011495479692174515376, 10.15457775335376934482094003497, 10.82625152067697145650880543079, 11.90644868723311745963233340624, 12.85997544459122992018468645490, 13.6289223044488268572515425861, 14.33699790683549475999422675042, 14.774337286256269561732128871303, 16.69731215643103102025252583909, 17.5059048356698034915828998076, 18.13248036414496338893960769826, 18.903771254878291141269301490954, 19.71855132109707557922784513430, 20.38742644260949226243843509030, 21.33027729564396736045795923577, 21.730042288499312273150444663596, 23.03684264477346968385674327412

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