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Properties

Label 1-569-569.219-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.723 + 0.690i$
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  + (0.325 − 0.945i)2-s + (0.487 − 0.873i)3-s + (−0.787 − 0.616i)4-s + (0.984 − 0.176i)5-s + (−0.666 − 0.745i)6-s + (−0.991 + 0.132i)7-s + (−0.839 + 0.544i)8-s + (−0.525 − 0.850i)9-s + (0.154 − 0.988i)10-s + (−0.598 − 0.801i)11-s + (−0.921 + 0.387i)12-s + (−0.883 − 0.467i)13-s + (−0.197 + 0.980i)14-s + (0.325 − 0.945i)15-s + (0.240 + 0.970i)16-s + (0.903 − 0.428i)17-s + ⋯
L(s)  = 1  + (0.325 − 0.945i)2-s + (0.487 − 0.873i)3-s + (−0.787 − 0.616i)4-s + (0.984 − 0.176i)5-s + (−0.666 − 0.745i)6-s + (−0.991 + 0.132i)7-s + (−0.839 + 0.544i)8-s + (−0.525 − 0.850i)9-s + (0.154 − 0.988i)10-s + (−0.598 − 0.801i)11-s + (−0.921 + 0.387i)12-s + (−0.883 − 0.467i)13-s + (−0.197 + 0.980i)14-s + (0.325 − 0.945i)15-s + (0.240 + 0.970i)16-s + (0.903 − 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.723 + 0.690i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.723 + 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4613405263 - 1.151022978i\)
\(L(\frac12)\) \(\approx\) \(-0.4613405263 - 1.151022978i\)
\(L(1)\) \(\approx\) \(0.5938794878 - 0.9888770047i\)
\(L(1)\) \(\approx\) \(0.5938794878 - 0.9888770047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.325 - 0.945i)T \)
3 \( 1 + (0.487 - 0.873i)T \)
5 \( 1 + (0.984 - 0.176i)T \)
7 \( 1 + (-0.991 + 0.132i)T \)
11 \( 1 + (-0.598 - 0.801i)T \)
13 \( 1 + (-0.883 - 0.467i)T \)
17 \( 1 + (0.903 - 0.428i)T \)
19 \( 1 + (-0.787 + 0.616i)T \)
23 \( 1 + (-0.367 - 0.930i)T \)
29 \( 1 + (0.903 + 0.428i)T \)
31 \( 1 + (0.408 + 0.912i)T \)
37 \( 1 + (-0.991 + 0.132i)T \)
41 \( 1 + (-0.367 - 0.930i)T \)
43 \( 1 + (0.325 + 0.945i)T \)
47 \( 1 + (-0.975 - 0.219i)T \)
53 \( 1 + (-0.666 - 0.745i)T \)
59 \( 1 + (-0.975 - 0.219i)T \)
61 \( 1 + (0.862 - 0.506i)T \)
67 \( 1 + (0.903 + 0.428i)T \)
71 \( 1 + (-0.839 - 0.544i)T \)
73 \( 1 + (-0.787 + 0.616i)T \)
79 \( 1 + (-0.110 - 0.993i)T \)
83 \( 1 + (0.964 - 0.262i)T \)
89 \( 1 + (0.408 - 0.912i)T \)
97 \( 1 + (0.937 - 0.346i)T \)
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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

−23.69662223947160469561012176270, −22.876099462929338973107829421163, −22.067697427801298677616906187779, −21.514430655675593718716691727237, −20.803218880094603306937622008887, −19.5692159235085233721184077706, −18.76958137211404278603397545537, −17.40949091336132812013420510818, −17.07083401634942443756211924629, −16.08198181934552908876795440981, −15.31023436577401538434265802046, −14.61903026459625253861394468178, −13.74582446783597459306972039822, −13.11440972839543551103630258659, −12.13310188909465450108461689572, −10.390496222332650448244120384, −9.77252968444616518190367380731, −9.25457263273539985176075604291, −8.05645425139285425229701263748, −7.06448460354616195693062910163, −6.10881499688771983455145267059, −5.18737154037828777399680139971, −4.34282028625096695761704078709, −3.20361078080764954272199918761, −2.30064644447744450848779785774, 0.50233168390733643490441013141, 1.798091896129123516649511633335, 2.79451354231380367350550398361, 3.281114802744905853163656205428, 4.995400597840260301376726104713, 5.90332326252530772587156447160, 6.6593332855051239747356486443, 8.18330090561347236096094588394, 8.938792716195262548397041444440, 9.980570258533282133742121172964, 10.44576820548231901110664778762, 12.03038794698871240387150324794, 12.616296545225185672921926031498, 13.16332311614912099165496274530, 14.11035828249855111560981108162, 14.53324112674691033837266614675, 16.00897540004739162275296397134, 17.171723615427901200625986343890, 18.01755267067251951182502567818, 18.86244767534393012752109785256, 19.29375783189344663647808848275, 20.288439264836057560624777968565, 21.01163068401410707083026140049, 21.77864109339361753494586926629, 22.697339837690445868305341983451

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