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Properties

Label 1-569-569.87-r0-0-0
Degree $1$
Conductor $569$
Sign $0.989 - 0.146i$
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  + (0.240 + 0.970i)2-s + (0.448 − 0.894i)3-s + (−0.883 + 0.467i)4-s + (0.759 − 0.650i)5-s + (0.975 + 0.219i)6-s + (0.862 − 0.506i)7-s + (−0.666 − 0.745i)8-s + (−0.598 − 0.801i)9-s + (0.814 + 0.580i)10-s + (0.839 + 0.544i)11-s + (0.0221 + 0.999i)12-s + (−0.367 + 0.930i)13-s + (0.699 + 0.714i)14-s + (−0.240 − 0.970i)15-s + (0.562 − 0.826i)16-s + (−0.197 − 0.980i)17-s + ⋯
L(s)  = 1  + (0.240 + 0.970i)2-s + (0.448 − 0.894i)3-s + (−0.883 + 0.467i)4-s + (0.759 − 0.650i)5-s + (0.975 + 0.219i)6-s + (0.862 − 0.506i)7-s + (−0.666 − 0.745i)8-s + (−0.598 − 0.801i)9-s + (0.814 + 0.580i)10-s + (0.839 + 0.544i)11-s + (0.0221 + 0.999i)12-s + (−0.367 + 0.930i)13-s + (0.699 + 0.714i)14-s + (−0.240 − 0.970i)15-s + (0.562 − 0.826i)16-s + (−0.197 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.989 - 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.068790424 - 0.1528488688i\)
\(L(\frac12)\) \(\approx\) \(2.068790424 - 0.1528488688i\)
\(L(1)\) \(\approx\) \(1.533984321 + 0.08296680153i\)
\(L(1)\) \(\approx\) \(1.533984321 + 0.08296680153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.240 + 0.970i)T \)
3 \( 1 + (0.448 - 0.894i)T \)
5 \( 1 + (0.759 - 0.650i)T \)
7 \( 1 + (0.862 - 0.506i)T \)
11 \( 1 + (0.839 + 0.544i)T \)
13 \( 1 + (-0.367 + 0.930i)T \)
17 \( 1 + (-0.197 - 0.980i)T \)
19 \( 1 + (0.883 + 0.467i)T \)
23 \( 1 + (-0.0663 + 0.997i)T \)
29 \( 1 + (0.197 - 0.980i)T \)
31 \( 1 + (0.110 + 0.993i)T \)
37 \( 1 + (-0.862 + 0.506i)T \)
41 \( 1 + (0.0663 - 0.997i)T \)
43 \( 1 + (0.240 - 0.970i)T \)
47 \( 1 + (-0.633 - 0.773i)T \)
53 \( 1 + (0.975 + 0.219i)T \)
59 \( 1 + (-0.633 - 0.773i)T \)
61 \( 1 + (-0.525 - 0.850i)T \)
67 \( 1 + (-0.197 + 0.980i)T \)
71 \( 1 + (-0.666 + 0.745i)T \)
73 \( 1 + (0.883 + 0.467i)T \)
79 \( 1 + (0.903 - 0.428i)T \)
83 \( 1 + (-0.487 + 0.873i)T \)
89 \( 1 + (0.110 - 0.993i)T \)
97 \( 1 + (-0.154 + 0.988i)T \)
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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

−22.5214335512378964060692875259, −22.38408983427011035585729647340, −21.47842602645965746483778349678, −21.049222289111611936507329196756, −20.02446521959262743881243501011, −19.41132397353791702921175287302, −18.274922479801147678125954172156, −17.6731062706987940552934910808, −16.710087734710930636456666431541, −15.19387552447628086560514586877, −14.6925190861809719296762838839, −14.0856478114436009184400040552, −13.18216462588735025579326433631, −11.99580038036731918751875158019, −10.98061264513842392880655228799, −10.576489433170848973429050701210, −9.550392730414103565089076890481, −8.875524883253613802403998299618, −7.949135634993703148268718530608, −6.135721547524295687341237797060, −5.331780056323190038227567892046, −4.42323015713851947958563414410, −3.25819561834819355681798002010, −2.586573662657840333653499311588, −1.490836504852151586185137336769, 1.09698323451086798663108709513, 2.05317406767219017082330333916, 3.680559484134084099732062028073, 4.755631820531226804564504814368, 5.58769806183084480284705731003, 6.809906819568588228020716143227, 7.27762832969862783462792090827, 8.33025177338200109195168014319, 9.14491941064289709198277872493, 9.81667055709910222559044637343, 11.80185381004594206666322865427, 12.18685054540979171441609699686, 13.54081042054335180475493344302, 13.90816683543104778463479463049, 14.4264807713572874484355520697, 15.59832256775256336762836177197, 16.765328823704455516416441496241, 17.40378771089481443787771246229, 17.87938409604637891441161101739, 18.852492091016485312648271270880, 19.994599539124733208988683251957, 20.761072297088653116279521055149, 21.59223408855819775383312472067, 22.69242517015336569402811425581, 23.51230978763957467339010537616

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