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Properties

Label 1-569-569.197-r0-0-0
Degree $1$
Conductor $569$
Sign $0.855 - 0.518i$
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  + (0.964 − 0.262i)2-s + (0.975 + 0.219i)3-s + (0.862 − 0.506i)4-s + (−0.883 + 0.467i)5-s + (0.999 − 0.0442i)6-s + (0.408 − 0.912i)7-s + (0.699 − 0.714i)8-s + (0.903 + 0.428i)9-s + (−0.730 + 0.683i)10-s + (0.197 − 0.980i)11-s + (0.952 − 0.304i)12-s + (−0.525 + 0.850i)13-s + (0.154 − 0.988i)14-s + (−0.964 + 0.262i)15-s + (0.487 − 0.873i)16-s + (0.937 + 0.346i)17-s + ⋯
L(s)  = 1  + (0.964 − 0.262i)2-s + (0.975 + 0.219i)3-s + (0.862 − 0.506i)4-s + (−0.883 + 0.467i)5-s + (0.999 − 0.0442i)6-s + (0.408 − 0.912i)7-s + (0.699 − 0.714i)8-s + (0.903 + 0.428i)9-s + (−0.730 + 0.683i)10-s + (0.197 − 0.980i)11-s + (0.952 − 0.304i)12-s + (−0.525 + 0.850i)13-s + (0.154 − 0.988i)14-s + (−0.964 + 0.262i)15-s + (0.487 − 0.873i)16-s + (0.937 + 0.346i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.855 - 0.518i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.855 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.182630117 - 0.8896334008i\)
\(L(\frac12)\) \(\approx\) \(3.182630117 - 0.8896334008i\)
\(L(1)\) \(\approx\) \(2.298571591 - 0.3908875898i\)
\(L(1)\) \(\approx\) \(2.298571591 - 0.3908875898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.964 - 0.262i)T \)
3 \( 1 + (0.975 + 0.219i)T \)
5 \( 1 + (-0.883 + 0.467i)T \)
7 \( 1 + (0.408 - 0.912i)T \)
11 \( 1 + (0.197 - 0.980i)T \)
13 \( 1 + (-0.525 + 0.850i)T \)
17 \( 1 + (0.937 + 0.346i)T \)
19 \( 1 + (-0.862 - 0.506i)T \)
23 \( 1 + (0.598 + 0.801i)T \)
29 \( 1 + (-0.937 + 0.346i)T \)
31 \( 1 + (0.0221 - 0.999i)T \)
37 \( 1 + (-0.408 + 0.912i)T \)
41 \( 1 + (-0.598 - 0.801i)T \)
43 \( 1 + (0.964 + 0.262i)T \)
47 \( 1 + (-0.984 + 0.176i)T \)
53 \( 1 + (0.999 - 0.0442i)T \)
59 \( 1 + (-0.984 + 0.176i)T \)
61 \( 1 + (-0.110 + 0.993i)T \)
67 \( 1 + (0.937 - 0.346i)T \)
71 \( 1 + (0.699 + 0.714i)T \)
73 \( 1 + (-0.862 - 0.506i)T \)
79 \( 1 + (0.996 + 0.0883i)T \)
83 \( 1 + (0.666 + 0.745i)T \)
89 \( 1 + (0.0221 + 0.999i)T \)
97 \( 1 + (-0.562 + 0.826i)T \)
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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

−23.25338925208019145147241202259, −22.75480103645852576977449775416, −21.519771269364402254647080275072, −20.81841450647106328035654008833, −20.2160415033504645891758396511, −19.41596824495234610555060967226, −18.5315982816470231331830942516, −17.330366750311418393306007824722, −16.26704130027079530149732621971, −15.32330443363245401459861977719, −14.89641291456493014899492970462, −14.320263485440141969395564441558, −12.83301644887251744790181356782, −12.53853700282392112543337714933, −11.854056534602960822745117743960, −10.53945400255215803247189176089, −9.234654330883381217454342736265, −8.21333221562378653438638976763, −7.698105580470863648857905081353, −6.7762093626603075236052782927, −5.359823661582016281143108024321, −4.59388787735505862912812235256, −3.59971851616319774458161551991, −2.659999445020213693343534508079, −1.65505912139744882887629015601, 1.32211366338685053721252152684, 2.59346515057164264497868757937, 3.65074889156616944876656041131, 4.01978001307293921909607752348, 5.09939001044974478402790343509, 6.63402260585103245367604524099, 7.3823831153849100627109438694, 8.153386980090046645981426347482, 9.44866891510646886950170133482, 10.586145591298873054937195539684, 11.15524181078781998033518976769, 12.095581858994273007686057937010, 13.27783996316855022559677272977, 13.89879099882220717625622236921, 14.733386023664737645630166348365, 15.12881107418283646150910730709, 16.28646810493499109146061177491, 16.924496144464748555463778054338, 18.85573939159610583256792199204, 19.19865594151690426501759275660, 19.892607773179833100994322573846, 20.815457205215263381959983579546, 21.43751463959077845553058884623, 22.238865411152926619329135520583, 23.2933030802794535254468493978

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