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Properties

Label 1-569-569.61-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.316 + 0.948i$
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  + (0.448 − 0.894i)2-s + (0.387 − 0.921i)3-s + (−0.598 − 0.801i)4-s + (0.0663 − 0.997i)5-s + (−0.650 − 0.759i)6-s + (−0.903 − 0.428i)7-s + (−0.984 + 0.176i)8-s + (−0.699 − 0.714i)9-s + (−0.862 − 0.506i)10-s + (−0.0442 + 0.999i)11-s + (−0.970 + 0.240i)12-s + (0.839 − 0.544i)13-s + (−0.787 + 0.616i)14-s + (−0.894 − 0.448i)15-s + (−0.283 + 0.958i)16-s + (−0.814 + 0.580i)17-s + ⋯
L(s)  = 1  + (0.448 − 0.894i)2-s + (0.387 − 0.921i)3-s + (−0.598 − 0.801i)4-s + (0.0663 − 0.997i)5-s + (−0.650 − 0.759i)6-s + (−0.903 − 0.428i)7-s + (−0.984 + 0.176i)8-s + (−0.699 − 0.714i)9-s + (−0.862 − 0.506i)10-s + (−0.0442 + 0.999i)11-s + (−0.970 + 0.240i)12-s + (0.839 − 0.544i)13-s + (−0.787 + 0.616i)14-s + (−0.894 − 0.448i)15-s + (−0.283 + 0.958i)16-s + (−0.814 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.316 + 0.948i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6609011967 - 0.9172949799i\)
\(L(\frac12)\) \(\approx\) \(-0.6609011967 - 0.9172949799i\)
\(L(1)\) \(\approx\) \(0.4476672591 - 1.001122085i\)
\(L(1)\) \(\approx\) \(0.4476672591 - 1.001122085i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.448 - 0.894i)T \)
3 \( 1 + (0.387 - 0.921i)T \)
5 \( 1 + (0.0663 - 0.997i)T \)
7 \( 1 + (-0.903 - 0.428i)T \)
11 \( 1 + (-0.0442 + 0.999i)T \)
13 \( 1 + (0.839 - 0.544i)T \)
17 \( 1 + (-0.814 + 0.580i)T \)
19 \( 1 + (-0.801 - 0.598i)T \)
23 \( 1 + (0.745 + 0.666i)T \)
29 \( 1 + (0.580 - 0.814i)T \)
31 \( 1 + (0.346 - 0.937i)T \)
37 \( 1 + (0.428 - 0.903i)T \)
41 \( 1 + (-0.666 + 0.745i)T \)
43 \( 1 + (-0.448 - 0.894i)T \)
47 \( 1 + (-0.304 + 0.952i)T \)
53 \( 1 + (0.650 + 0.759i)T \)
59 \( 1 + (0.304 - 0.952i)T \)
61 \( 1 + (0.197 - 0.980i)T \)
67 \( 1 + (-0.814 - 0.580i)T \)
71 \( 1 + (-0.984 - 0.176i)T \)
73 \( 1 + (0.801 + 0.598i)T \)
79 \( 1 + (-0.154 - 0.988i)T \)
83 \( 1 + (0.773 - 0.633i)T \)
89 \( 1 + (-0.346 - 0.937i)T \)
97 \( 1 + (-0.132 + 0.991i)T \)
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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

−23.66748113117478262293898624154, −22.836213500062667959461329480742, −22.269178180162720706302217226007, −21.55199143341075047687432709084, −20.9743467480048986301182986535, −19.540411054018218163524306400548, −18.76975000711693220033297400923, −17.99184561453113989415738997962, −16.56241732563801087994161611289, −16.27221778361135855657198088862, −15.34387788082635905663660598268, −14.743175807555670815373918264092, −13.79377659995817890921996471629, −13.30265206682168633859925104725, −11.86674956010483928074283237883, −10.88574641539347120288756277468, −9.97784752319502175123175790630, −8.787043393599542413697311697524, −8.48938220208547159775545071224, −6.86466343717804749299145939632, −6.34671081501391932839033794764, −5.35651900844599977694739420358, −4.14235705469860634182628924931, −3.2842558062563799509586163490, −2.670426685673570160948107816581, 0.468775042716761350267133256943, 1.57654304024197827866487309228, 2.53819958543123268097002298381, 3.71491248542472442176466386752, 4.58179830818825123965803815156, 5.88411169722006500368804679593, 6.64069022029112275590706931485, 7.98282803988000727015187420380, 8.95757498928216243422104490784, 9.61135655845768086373408913806, 10.73876666994491609838637161915, 11.818107961087777471117924789712, 12.72345886609551074290088427057, 13.183567609923620379580193468224, 13.56584685805804071391998083279, 14.984776513155274740777765247393, 15.65344558676290631576818134233, 17.22628025938906259807988303749, 17.65723872069623800702662854086, 18.846502297176386589860908252606, 19.58435920870482364362775756330, 20.142583271571524232707882689308, 20.71630301285104289774403716951, 21.73121831754736277251672572577, 23.01484535265620239793355436744

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