Properties

Label 1-571-571.394-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: $\chi_{571} (394, \cdot )$
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

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

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