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Properties

Label 1-569-569.55-r1-0-0
Degree $1$
Conductor $569$
Sign $-0.485 + 0.874i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (−0.262 − 0.964i)2-s + (−0.534 − 0.845i)3-s + (−0.862 + 0.506i)4-s + (−0.883 + 0.467i)5-s + (−0.675 + 0.737i)6-s + (−0.912 − 0.408i)7-s + (0.714 + 0.699i)8-s + (−0.428 + 0.903i)9-s + (0.683 + 0.730i)10-s + (0.553 + 0.833i)11-s + (0.888 + 0.457i)12-s + (0.850 + 0.525i)13-s + (−0.154 + 0.988i)14-s + (0.867 + 0.496i)15-s + (0.487 − 0.873i)16-s + (0.346 − 0.937i)17-s + ⋯
L(s)  = 1  + (−0.262 − 0.964i)2-s + (−0.534 − 0.845i)3-s + (−0.862 + 0.506i)4-s + (−0.883 + 0.467i)5-s + (−0.675 + 0.737i)6-s + (−0.912 − 0.408i)7-s + (0.714 + 0.699i)8-s + (−0.428 + 0.903i)9-s + (0.683 + 0.730i)10-s + (0.553 + 0.833i)11-s + (0.888 + 0.457i)12-s + (0.850 + 0.525i)13-s + (−0.154 + 0.988i)14-s + (0.867 + 0.496i)15-s + (0.487 − 0.873i)16-s + (0.346 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.485 + 0.874i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ -0.485 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01581781988 + 0.02687617112i\)
\(L(\frac12)\) \(\approx\) \(0.01581781988 + 0.02687617112i\)
\(L(1)\) \(\approx\) \(0.4580071430 - 0.2573157566i\)
\(L(1)\) \(\approx\) \(0.4580071430 - 0.2573157566i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.262 - 0.964i)T \)
3 \( 1 + (-0.534 - 0.845i)T \)
5 \( 1 + (-0.883 + 0.467i)T \)
7 \( 1 + (-0.912 - 0.408i)T \)
11 \( 1 + (0.553 + 0.833i)T \)
13 \( 1 + (0.850 + 0.525i)T \)
17 \( 1 + (0.346 - 0.937i)T \)
19 \( 1 + (0.251 + 0.967i)T \)
23 \( 1 + (-0.989 - 0.143i)T \)
29 \( 1 + (-0.418 + 0.908i)T \)
31 \( 1 + (-0.691 - 0.722i)T \)
37 \( 1 + (-0.356 - 0.934i)T \)
41 \( 1 + (0.598 + 0.801i)T \)
43 \( 1 + (-0.964 - 0.262i)T \)
47 \( 1 + (0.571 - 0.820i)T \)
53 \( 1 + (0.737 + 0.675i)T \)
59 \( 1 + (0.820 + 0.571i)T \)
61 \( 1 + (0.993 + 0.110i)T \)
67 \( 1 + (0.346 + 0.937i)T \)
71 \( 1 + (-0.714 + 0.699i)T \)
73 \( 1 + (0.967 - 0.251i)T \)
79 \( 1 + (0.0883 - 0.996i)T \)
83 \( 1 + (-0.998 - 0.0552i)T \)
89 \( 1 + (0.722 + 0.691i)T \)
97 \( 1 + (-0.186 - 0.982i)T \)
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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

−23.72855246708970568553000201526, −22.76288720066763564117856542934, −22.26639280388693597223145721242, −21.38536315177808661677735714356, −20.07855139638927778096869070946, −19.38599203421549259856145852488, −18.51519639204854072376318044474, −17.42534585014484246921382832147, −16.66660765834780047747654783262, −15.92316208183142172168428438336, −15.615173266751403112079307663033, −14.723105774281968848483071337752, −13.50449039847448902740091216843, −12.56798654193471519552974966727, −11.541911660172853606661205002006, −10.59451666857361835199418596479, −9.584141056776722444726501696927, −8.78688558704053093568036657103, −8.1535446437385087006623269654, −6.73678479179656751941802322405, −5.951000447355338654578288765455, −5.22869500939630915657566980422, −3.924882272275632956750246752354, −3.49612039728725092447776794948, −0.89807342338620219173501635908, 0.01397106004215491961231465593, 1.12052214275338902793617354410, 2.277044353524301216067273340, 3.530129686829132419545498541159, 4.194307747141641484655029255371, 5.68176768687432345084936800678, 6.94787866391553857955258873407, 7.46770541961760558290887140845, 8.57893919837358583812195339844, 9.72924298540163015955996616483, 10.58352536206047525048055728515, 11.5280992206172638595379795169, 12.06865978963252447508152354389, 12.81010313678825330638471969819, 13.76379669906959447769828329879, 14.5490515796629687280981100446, 16.25358272188931902698782581174, 16.55086289059600214906572633751, 17.87419109376748720093641339927, 18.51503527582142411265102845635, 19.046738109308125489173528592, 20.008470182696255738808268114760, 20.37161544639637929044547988748, 21.91432835344617604394609725569, 22.65177193431988263558345461969

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