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Properties

Label 1-569-569.136-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.666 + 0.745i$
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  + (−0.283 − 0.958i)2-s + (−0.0221 − 0.999i)3-s + (−0.839 + 0.544i)4-s + (0.964 − 0.262i)5-s + (−0.952 + 0.304i)6-s + (−0.197 − 0.980i)7-s + (0.759 + 0.650i)8-s + (−0.999 + 0.0442i)9-s + (−0.525 − 0.850i)10-s + (0.984 − 0.176i)11-s + (0.562 + 0.826i)12-s + (−0.666 + 0.745i)13-s + (−0.883 + 0.467i)14-s + (−0.283 − 0.958i)15-s + (0.408 − 0.912i)16-s + (−0.787 + 0.616i)17-s + ⋯
L(s)  = 1  + (−0.283 − 0.958i)2-s + (−0.0221 − 0.999i)3-s + (−0.839 + 0.544i)4-s + (0.964 − 0.262i)5-s + (−0.952 + 0.304i)6-s + (−0.197 − 0.980i)7-s + (0.759 + 0.650i)8-s + (−0.999 + 0.0442i)9-s + (−0.525 − 0.850i)10-s + (0.984 − 0.176i)11-s + (0.562 + 0.826i)12-s + (−0.666 + 0.745i)13-s + (−0.883 + 0.467i)14-s + (−0.283 − 0.958i)15-s + (0.408 − 0.912i)16-s + (−0.787 + 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.666 + 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3227244381 - 0.7212900959i\)
\(L(\frac12)\) \(\approx\) \(-0.3227244381 - 0.7212900959i\)
\(L(1)\) \(\approx\) \(0.4473775222 - 0.6838008182i\)
\(L(1)\) \(\approx\) \(0.4473775222 - 0.6838008182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.283 - 0.958i)T \)
3 \( 1 + (-0.0221 - 0.999i)T \)
5 \( 1 + (0.964 - 0.262i)T \)
7 \( 1 + (-0.197 - 0.980i)T \)
11 \( 1 + (0.984 - 0.176i)T \)
13 \( 1 + (-0.666 + 0.745i)T \)
17 \( 1 + (-0.787 + 0.616i)T \)
19 \( 1 + (-0.839 - 0.544i)T \)
23 \( 1 + (-0.975 - 0.219i)T \)
29 \( 1 + (-0.787 - 0.616i)T \)
31 \( 1 + (0.154 - 0.988i)T \)
37 \( 1 + (-0.197 - 0.980i)T \)
41 \( 1 + (-0.975 - 0.219i)T \)
43 \( 1 + (-0.283 + 0.958i)T \)
47 \( 1 + (0.325 - 0.945i)T \)
53 \( 1 + (-0.952 + 0.304i)T \)
59 \( 1 + (0.325 - 0.945i)T \)
61 \( 1 + (0.699 - 0.714i)T \)
67 \( 1 + (-0.787 - 0.616i)T \)
71 \( 1 + (0.759 - 0.650i)T \)
73 \( 1 + (-0.839 - 0.544i)T \)
79 \( 1 + (0.814 + 0.580i)T \)
83 \( 1 + (-0.921 + 0.387i)T \)
89 \( 1 + (0.154 + 0.988i)T \)
97 \( 1 + (0.862 - 0.506i)T \)
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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

−23.95170823467997533365669332783, −22.54050021733697468879260268826, −22.27428093693028872595951102631, −21.72085078641765248604431959097, −20.49321021840145701807008166676, −19.562160425430589870010959601853, −18.51296618403049709212529501766, −17.63195884447070848176579340770, −17.114189574899412666297632605607, −16.19217443965592130281120482352, −15.33344206221791887435639057572, −14.69468308584037299824189757961, −14.07187930139659442196656711557, −12.925027986639023313444321139479, −11.76656866752594730552203811856, −10.426323434454965948541275778478, −9.84777826769237460004216769354, −9.05331566469480664498713819065, −8.46777286084593105194677766815, −6.94059570006384001863980909536, −6.06407594181386295336340883812, −5.39636188101922618203421933549, −4.50597162497761945778964817323, −3.185110622998104169624193888142, −1.87804628253796113879464263144, 0.42745115126041778882610466440, 1.76893809446995140831699463753, 2.194671174337196540324585671813, 3.72893005569758141639586896293, 4.6674332350153555156635234974, 6.14625133346937641036795719796, 6.875231583577938529511204896440, 8.04180551200415903256803852449, 9.01571621255264366169895562474, 9.71751222846917957436961780736, 10.79044766558503114163150524208, 11.59321297795132585329308171956, 12.5716894535748545022270519517, 13.264193021279256059680000177211, 13.883583949466852845383913937551, 14.5900849157845351960417784145, 16.72276201544424635240301263885, 17.09873545153135037513349171657, 17.6630087195060754712596079687, 18.69444930633588737215583444482, 19.55364587575819487856472833002, 19.97776811554706568628123212592, 20.91826055066344406780661838294, 21.984394340191883460693742799574, 22.42827348423741817578573621026

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