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Properties

Label 1-569-569.4-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.311 + 0.950i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (0.562 − 0.826i)2-s + (0.283 − 0.958i)3-s + (−0.367 − 0.930i)4-s + (−0.952 − 0.304i)5-s + (−0.633 − 0.773i)6-s + (−0.525 − 0.850i)7-s + (−0.975 − 0.219i)8-s + (−0.839 − 0.544i)9-s + (−0.787 + 0.616i)10-s + (0.666 − 0.745i)11-s + (−0.996 + 0.0883i)12-s + (0.0663 + 0.997i)13-s + (−0.999 − 0.0442i)14-s + (−0.562 + 0.826i)15-s + (−0.730 + 0.683i)16-s + (0.699 − 0.714i)17-s + ⋯
L(s)  = 1  + (0.562 − 0.826i)2-s + (0.283 − 0.958i)3-s + (−0.367 − 0.930i)4-s + (−0.952 − 0.304i)5-s + (−0.633 − 0.773i)6-s + (−0.525 − 0.850i)7-s + (−0.975 − 0.219i)8-s + (−0.839 − 0.544i)9-s + (−0.787 + 0.616i)10-s + (0.666 − 0.745i)11-s + (−0.996 + 0.0883i)12-s + (0.0663 + 0.997i)13-s + (−0.999 − 0.0442i)14-s + (−0.562 + 0.826i)15-s + (−0.730 + 0.683i)16-s + (0.699 − 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.311 + 0.950i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.311 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6440894210 - 0.8890244454i\)
\(L(\frac12)\) \(\approx\) \(-0.6440894210 - 0.8890244454i\)
\(L(1)\) \(\approx\) \(0.4430499318 - 0.9676975250i\)
\(L(1)\) \(\approx\) \(0.4430499318 - 0.9676975250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.562 - 0.826i)T \)
3 \( 1 + (0.283 - 0.958i)T \)
5 \( 1 + (-0.952 - 0.304i)T \)
7 \( 1 + (-0.525 - 0.850i)T \)
11 \( 1 + (0.666 - 0.745i)T \)
13 \( 1 + (0.0663 + 0.997i)T \)
17 \( 1 + (0.699 - 0.714i)T \)
19 \( 1 + (0.367 - 0.930i)T \)
23 \( 1 + (-0.964 - 0.262i)T \)
29 \( 1 + (-0.699 - 0.714i)T \)
31 \( 1 + (-0.903 + 0.428i)T \)
37 \( 1 + (0.525 + 0.850i)T \)
41 \( 1 + (0.964 + 0.262i)T \)
43 \( 1 + (0.562 + 0.826i)T \)
47 \( 1 + (0.921 + 0.387i)T \)
53 \( 1 + (-0.633 - 0.773i)T \)
59 \( 1 + (0.921 + 0.387i)T \)
61 \( 1 + (-0.598 - 0.801i)T \)
67 \( 1 + (0.699 + 0.714i)T \)
71 \( 1 + (-0.975 + 0.219i)T \)
73 \( 1 + (0.367 - 0.930i)T \)
79 \( 1 + (-0.197 - 0.980i)T \)
83 \( 1 + (0.448 - 0.894i)T \)
89 \( 1 + (-0.903 - 0.428i)T \)
97 \( 1 + (-0.814 - 0.580i)T \)
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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

−23.68729393844630355671705883229, −22.76357065111270553392370897638, −22.38909956005901204316705479915, −21.70447159469228758714710999987, −20.55937620638966007909419756012, −19.90049311884046483138084618410, −18.81704735494984959724430721279, −17.83749362734563246793614840441, −16.69728825206889031007308098767, −16.076767640525106974882135760724, −15.296162192471276499904662626740, −14.84075627794484850004282866597, −14.130189014760975812097691790696, −12.578787136410272898620897896652, −12.233995864009668297629669237761, −11.04761245437465099061937756144, −9.870528985492127965414024511619, −9.00127821549514533318885020516, −8.06128885704628363164332871082, −7.374405185419503820515390456198, −5.94254959829040175120351192461, −5.39943562092789699079529725587, −3.90889034135190482815264873985, −3.72048969585165201525609607938, −2.52830420113152378034209934180, 0.478619709505390133356068727037, 1.38643209822915860620565202452, 2.836869605769105299329178613864, 3.665622170652905506233970519206, 4.46756535308076629846028300215, 5.909653361560832195467509760927, 6.82863732129483240998731742873, 7.7090027409508231829922619348, 8.92098599880268009709959151295, 9.615500101504653810473574563238, 11.16662571439343855295340538373, 11.56361593734541485416991790973, 12.42194962571733116089917468005, 13.26558668592787790273511250623, 14.00529986804263499773437954779, 14.59086105825065193503527733957, 15.97019422538845536836645441074, 16.718966504728314825092433351431, 17.977940985304656680376798534561, 19.06897935288281432598829082254, 19.27547949640574805242207363117, 20.15249926699467314207887149617, 20.66396414523010680241926909206, 21.991375735539606702961013459058, 22.76907623671090497976219150968

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