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Properties

Label 1-571-571.100-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.257 - 0.966i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.451 + 0.892i)2-s + (−0.782 + 0.622i)3-s + (−0.592 + 0.805i)4-s + (0.528 − 0.849i)5-s + (−0.909 − 0.416i)6-s + (0.180 − 0.983i)7-s + (−0.986 − 0.164i)8-s + (0.224 − 0.974i)9-s + (0.996 + 0.0880i)10-s + (−0.421 + 0.906i)11-s + (−0.0385 − 0.999i)12-s + (−0.480 + 0.876i)13-s + (0.959 − 0.282i)14-s + (0.115 + 0.993i)15-s + (−0.298 − 0.954i)16-s + (−0.319 + 0.947i)17-s + ⋯
L(s)  = 1  + (0.451 + 0.892i)2-s + (−0.782 + 0.622i)3-s + (−0.592 + 0.805i)4-s + (0.528 − 0.849i)5-s + (−0.909 − 0.416i)6-s + (0.180 − 0.983i)7-s + (−0.986 − 0.164i)8-s + (0.224 − 0.974i)9-s + (0.996 + 0.0880i)10-s + (−0.421 + 0.906i)11-s + (−0.0385 − 0.999i)12-s + (−0.480 + 0.876i)13-s + (0.959 − 0.282i)14-s + (0.115 + 0.993i)15-s + (−0.298 − 0.954i)16-s + (−0.319 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.257 - 0.966i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.257 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005446797084 + 0.007084922512i\)
\(L(\frac12)\) \(\approx\) \(0.005446797084 + 0.007084922512i\)
\(L(1)\) \(\approx\) \(0.6406916921 + 0.3603286915i\)
\(L(1)\) \(\approx\) \(0.6406916921 + 0.3603286915i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.451 + 0.892i)T \)
3 \( 1 + (-0.782 + 0.622i)T \)
5 \( 1 + (0.528 - 0.849i)T \)
7 \( 1 + (0.180 - 0.983i)T \)
11 \( 1 + (-0.421 + 0.906i)T \)
13 \( 1 + (-0.480 + 0.876i)T \)
17 \( 1 + (-0.319 + 0.947i)T \)
19 \( 1 + (-0.834 - 0.551i)T \)
23 \( 1 + (-0.461 - 0.887i)T \)
29 \( 1 + (-0.821 - 0.569i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (-0.982 + 0.186i)T \)
41 \( 1 + (-0.926 + 0.376i)T \)
43 \( 1 + (-0.857 + 0.514i)T \)
47 \( 1 + (0.975 + 0.218i)T \)
53 \( 1 + (-0.889 + 0.456i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.256 - 0.966i)T \)
67 \( 1 + (0.840 + 0.542i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.329 - 0.944i)T \)
79 \( 1 + (0.411 + 0.911i)T \)
83 \( 1 + (0.0935 + 0.995i)T \)
89 \( 1 + (0.815 - 0.578i)T \)
97 \( 1 + (0.00551 - 0.999i)T \)
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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

−22.39257776954308119501867281298, −21.98555112967855007924224798228, −21.32731994312103666292399875413, −20.19701097259371202394806359793, −18.96505718831279186115314792401, −18.64955736834901512579069273208, −17.983821296795013025485399495981, −17.14074632937088347234464641361, −15.72424917959971724396614659893, −14.88771353916505857611213676659, −13.87500264488521129194249954445, −13.205516651224983296324523586075, −12.32425854720900905564084165126, −11.515512565416887201728129811675, −10.83379632663640438775524823123, −10.08740332526055866418122130829, −8.94900156098603221831601588498, −7.720959795931680777958616552108, −6.47728547489413974974698215562, −5.52956577503060940494089923695, −5.281772873832707704838655786044, −3.49011086929492811209326326179, −2.47828277911808543435564749298, −1.77309806212301486453526112070, −0.00405190954490111890255113132, 1.86400026900306759265896282028, 3.8210132975450978138427710969, 4.601753689149789881337841003531, 5.00721300686421495326204517722, 6.2819459479915841973489073347, 6.89401434838979096319855605739, 8.11626268676354262577795697262, 9.156986346353698682576512673883, 9.95861951209603997343897148646, 10.92069613969521855064564596700, 12.254527164036861247764209519239, 12.756710871651379357220976990948, 13.73542111251778737625110693235, 14.692932811221062529554342523779, 15.50000527070230124826081705895, 16.44670675232359449892948991085, 17.16486809344736143840167492753, 17.31274114223439101351742362065, 18.41635888210587457599041158647, 20.02489724396520939267819538294, 20.8042414817901582104854311840, 21.53171901009371009885271869921, 22.19060000893304506273793628001, 23.24418386717032523759018165782, 23.800420004252901742165007895307

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