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Properties

Label 1-569-569.420-r1-0-0
Degree $1$
Conductor $569$
Sign $0.999 - 0.0425i$
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  + (0.580 − 0.814i)2-s + (−0.251 − 0.967i)3-s + (−0.325 − 0.945i)4-s + (0.996 + 0.0883i)5-s + (−0.934 − 0.356i)6-s + (−0.997 − 0.0663i)7-s + (−0.958 − 0.283i)8-s + (−0.873 + 0.487i)9-s + (0.650 − 0.759i)10-s + (0.948 + 0.315i)11-s + (−0.833 + 0.553i)12-s + (0.970 − 0.240i)13-s + (−0.633 + 0.773i)14-s + (−0.165 − 0.986i)15-s + (−0.787 + 0.616i)16-s + (−0.219 + 0.975i)17-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.251 − 0.967i)3-s + (−0.325 − 0.945i)4-s + (0.996 + 0.0883i)5-s + (−0.934 − 0.356i)6-s + (−0.997 − 0.0663i)7-s + (−0.958 − 0.283i)8-s + (−0.873 + 0.487i)9-s + (0.650 − 0.759i)10-s + (0.948 + 0.315i)11-s + (−0.833 + 0.553i)12-s + (0.970 − 0.240i)13-s + (−0.633 + 0.773i)14-s + (−0.165 − 0.986i)15-s + (−0.787 + 0.616i)16-s + (−0.219 + 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.999 - 0.0425i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (420, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.999 - 0.0425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.434664801 + 0.03055087413i\)
\(L(\frac12)\) \(\approx\) \(1.434664801 + 0.03055087413i\)
\(L(1)\) \(\approx\) \(0.9844076221 - 0.6694334245i\)
\(L(1)\) \(\approx\) \(0.9844076221 - 0.6694334245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.580 - 0.814i)T \)
3 \( 1 + (-0.251 - 0.967i)T \)
5 \( 1 + (0.996 + 0.0883i)T \)
7 \( 1 + (-0.997 - 0.0663i)T \)
11 \( 1 + (0.948 + 0.315i)T \)
13 \( 1 + (0.970 - 0.240i)T \)
17 \( 1 + (-0.219 + 0.975i)T \)
19 \( 1 + (-0.898 + 0.438i)T \)
23 \( 1 + (-0.982 - 0.186i)T \)
29 \( 1 + (-0.534 + 0.845i)T \)
31 \( 1 + (-0.208 + 0.977i)T \)
37 \( 1 + (0.658 + 0.752i)T \)
41 \( 1 + (-0.562 - 0.826i)T \)
43 \( 1 + (-0.814 + 0.580i)T \)
47 \( 1 + (0.780 + 0.624i)T \)
53 \( 1 + (-0.356 + 0.934i)T \)
59 \( 1 + (-0.624 + 0.780i)T \)
61 \( 1 + (0.262 - 0.964i)T \)
67 \( 1 + (-0.219 - 0.975i)T \)
71 \( 1 + (0.958 - 0.283i)T \)
73 \( 1 + (0.438 + 0.898i)T \)
79 \( 1 + (-0.745 + 0.666i)T \)
83 \( 1 + (0.794 - 0.607i)T \)
89 \( 1 + (-0.977 + 0.208i)T \)
97 \( 1 + (-0.820 + 0.571i)T \)
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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

−22.88384814871212497862010729902, −22.15578394826220858387449806833, −21.76745687470226061818369269438, −20.86497385901156873538143535260, −20.06660197195259521763714595604, −18.638207287947718683193295885583, −17.69120095168152297420358508867, −16.79406567710520466897048571532, −16.41110147090311644356996752483, −15.54427988704982579884425841284, −14.67930848073770549157143351027, −13.724343953050517494653160801120, −13.235555725734737493020349835351, −11.98970243569585093312265638254, −11.12072754973293951938710346975, −9.76384485276816704324010605019, −9.26386615839619506337558700984, −8.46297873511159054363231038740, −6.75662362277490605066394502938, −6.1663572604555704389229260906, −5.53577912748938037447360966986, −4.288205675371200533126661637805, −3.60384830874173621667613983561, −2.43330345767962431485032553904, −0.29164824278668903594227653189, 1.2566428125290985822076636286, 1.87205077186367878681663346783, 3.041194698765087050617996587, 4.05198845281597807843259099117, 5.54375520906047476190922293118, 6.31716853569864474259657367879, 6.643777160514648628739768961294, 8.467892930825585319960248701821, 9.33054194880245818117323329383, 10.37567481760986222240715100741, 11.00635309860933436572534239187, 12.38285752250501398228503190331, 12.6458975502125867062716474064, 13.57239330850798245475997360274, 14.13333091781528822388431567516, 15.12997924547366839767546241862, 16.54186166877757615190044122552, 17.36588159836651013567186880746, 18.30015061657222480807792021703, 18.876833224642061816302937192153, 19.8317989862134625579196117562, 20.35163552385699868725075230127, 21.67556251837557024113010846902, 22.14777403432059828694611943232, 22.9816878283976865860919085789

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