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Properties

Label 1-571-571.68-r1-0-0
Degree $1$
Conductor $571$
Sign $0.611 + 0.791i$
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  + (0.879 + 0.475i)2-s + (−0.652 − 0.757i)3-s + (0.546 + 0.837i)4-s + (−0.0165 − 0.999i)5-s + (−0.213 − 0.976i)6-s + (0.768 + 0.639i)7-s + (0.0825 + 0.996i)8-s + (−0.148 + 0.988i)9-s + (0.461 − 0.887i)10-s + (−0.461 − 0.887i)11-s + (0.277 − 0.960i)12-s + (0.490 + 0.871i)13-s + (0.371 + 0.928i)14-s + (−0.746 + 0.665i)15-s + (−0.401 + 0.915i)16-s + (0.995 + 0.0990i)17-s + ⋯
L(s)  = 1  + (0.879 + 0.475i)2-s + (−0.652 − 0.757i)3-s + (0.546 + 0.837i)4-s + (−0.0165 − 0.999i)5-s + (−0.213 − 0.976i)6-s + (0.768 + 0.639i)7-s + (0.0825 + 0.996i)8-s + (−0.148 + 0.988i)9-s + (0.461 − 0.887i)10-s + (−0.461 − 0.887i)11-s + (0.277 − 0.960i)12-s + (0.490 + 0.871i)13-s + (0.371 + 0.928i)14-s + (−0.746 + 0.665i)15-s + (−0.401 + 0.915i)16-s + (0.995 + 0.0990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.700860822 + 1.326969082i\)
\(L(\frac12)\) \(\approx\) \(2.700860822 + 1.326969082i\)
\(L(1)\) \(\approx\) \(1.588922226 + 0.2461061981i\)
\(L(1)\) \(\approx\) \(1.588922226 + 0.2461061981i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.879 + 0.475i)T \)
3 \( 1 + (-0.652 - 0.757i)T \)
5 \( 1 + (-0.0165 - 0.999i)T \)
7 \( 1 + (0.768 + 0.639i)T \)
11 \( 1 + (-0.461 - 0.887i)T \)
13 \( 1 + (0.490 + 0.871i)T \)
17 \( 1 + (0.995 + 0.0990i)T \)
19 \( 1 + (0.973 + 0.229i)T \)
23 \( 1 + (-0.518 - 0.854i)T \)
29 \( 1 + (-0.677 + 0.735i)T \)
31 \( 1 + (-0.401 + 0.915i)T \)
37 \( 1 + (-0.909 + 0.416i)T \)
41 \( 1 + (-0.945 + 0.324i)T \)
43 \( 1 + (0.701 + 0.712i)T \)
47 \( 1 + (0.401 - 0.915i)T \)
53 \( 1 + (0.724 + 0.689i)T \)
59 \( 1 + (0.245 + 0.969i)T \)
61 \( 1 + (0.991 - 0.131i)T \)
67 \( 1 + (0.724 + 0.689i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.909 - 0.416i)T \)
79 \( 1 + (-0.0495 - 0.998i)T \)
83 \( 1 + (-0.213 - 0.976i)T \)
89 \( 1 + (0.213 + 0.976i)T \)
97 \( 1 + (0.965 - 0.261i)T \)
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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

−22.68652782840723939536058773376, −22.3650178591830957406925530522, −21.22313228059567748447516571266, −20.68789031836688556560525314986, −20.020605624951096871910428752227, −18.66848654327620936578427610377, −17.936848403079488363341261835647, −17.1115399073378834014964393595, −15.71522428179285304564929270299, −15.35462905781673770098542808621, −14.4370140848968028628570009096, −13.7463057392629689339338396838, −12.55689080457911001959226487498, −11.54065887795913592652026704388, −11.06944995338231939950955007120, −10.13605053936326626983336713317, −9.77899886975027256042444618977, −7.736808843278610793914068999365, −7.004903945100066971948793777127, −5.67273555264115958407510683202, −5.24388333439326492703667214300, −3.945393987086877196474726917302, −3.42399012875899993943958585103, −2.06357853974509721801613339893, −0.64957293767384985636000158570, 1.1345395469480156463291375779, 2.09801212883326765675645967743, 3.545722371171480833013552613758, 4.90197664917133664671658239109, 5.41179643880152553380383901026, 6.129402934815114849089557201752, 7.345158985447508217389924344109, 8.21556392411557309895776296430, 8.80620251379321312589899517969, 10.646027466186712744730209457594, 11.74120142705868085122234895632, 12.01998981069939447753593117607, 12.947593420987663617261106931715, 13.80036345593428178729342567687, 14.42349838626411601095747102134, 15.85873850463975601379547704088, 16.38743955408212695170314434386, 17.00634247501847634448032355071, 18.12974121693609883745665008040, 18.73758180696217303879871297527, 20.04778470216627855475766117515, 20.98885375744447853254695658850, 21.542883059407563111736933806739, 22.41128294316885191370235293369, 23.5648549408759118303589445348

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