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Properties

Label 1-569-569.90-r0-0-0
Degree $1$
Conductor $569$
Sign $0.999 + 0.0406i$
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  + (−0.999 + 0.0442i)2-s + (−0.883 + 0.467i)3-s + (0.996 − 0.0883i)4-s + (0.903 + 0.428i)5-s + (0.862 − 0.506i)6-s + (0.325 − 0.945i)7-s + (−0.991 + 0.132i)8-s + (0.562 − 0.826i)9-s + (−0.921 − 0.387i)10-s + (−0.730 + 0.683i)11-s + (−0.839 + 0.544i)12-s + (0.937 + 0.346i)13-s + (−0.283 + 0.958i)14-s + (−0.999 + 0.0442i)15-s + (0.984 − 0.176i)16-s + (−0.448 − 0.894i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0442i)2-s + (−0.883 + 0.467i)3-s + (0.996 − 0.0883i)4-s + (0.903 + 0.428i)5-s + (0.862 − 0.506i)6-s + (0.325 − 0.945i)7-s + (−0.991 + 0.132i)8-s + (0.562 − 0.826i)9-s + (−0.921 − 0.387i)10-s + (−0.730 + 0.683i)11-s + (−0.839 + 0.544i)12-s + (0.937 + 0.346i)13-s + (−0.283 + 0.958i)14-s + (−0.999 + 0.0442i)15-s + (0.984 − 0.176i)16-s + (−0.448 − 0.894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.999 + 0.0406i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.999 + 0.0406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8175198352 + 0.01663302860i\)
\(L(\frac12)\) \(\approx\) \(0.8175198352 + 0.01663302860i\)
\(L(1)\) \(\approx\) \(0.6855156469 + 0.05801844125i\)
\(L(1)\) \(\approx\) \(0.6855156469 + 0.05801844125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.999 + 0.0442i)T \)
3 \( 1 + (-0.883 + 0.467i)T \)
5 \( 1 + (0.903 + 0.428i)T \)
7 \( 1 + (0.325 - 0.945i)T \)
11 \( 1 + (-0.730 + 0.683i)T \)
13 \( 1 + (0.937 + 0.346i)T \)
17 \( 1 + (-0.448 - 0.894i)T \)
19 \( 1 + (0.996 + 0.0883i)T \)
23 \( 1 + (0.154 - 0.988i)T \)
29 \( 1 + (-0.448 + 0.894i)T \)
31 \( 1 + (0.964 + 0.262i)T \)
37 \( 1 + (0.325 - 0.945i)T \)
41 \( 1 + (0.154 - 0.988i)T \)
43 \( 1 + (-0.999 - 0.0442i)T \)
47 \( 1 + (-0.525 - 0.850i)T \)
53 \( 1 + (0.862 - 0.506i)T \)
59 \( 1 + (-0.525 - 0.850i)T \)
61 \( 1 + (0.240 + 0.970i)T \)
67 \( 1 + (-0.448 + 0.894i)T \)
71 \( 1 + (-0.991 - 0.132i)T \)
73 \( 1 + (0.996 + 0.0883i)T \)
79 \( 1 + (0.487 + 0.873i)T \)
83 \( 1 + (-0.787 - 0.616i)T \)
89 \( 1 + (0.964 - 0.262i)T \)
97 \( 1 + (0.633 + 0.773i)T \)
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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

−23.53394320554986334576147099408, −22.214247529206124324402866790025, −21.403854894810584729129548039951, −20.94697223044142394704832837116, −19.714645717417994225239606554245, −18.64984137029716909069102552080, −18.22845381103752477805911999585, −17.55300643878439263336819877731, −16.798898792767955654670505371538, −15.89919904618912243860140783110, −15.25715790450623900996157385300, −13.56302889834858857758836174945, −12.99680962567732236724552185414, −11.85256096309141388496587089120, −11.24261252214392200946642171666, −10.33937471851751730822469835314, −9.443934355462773058342266222503, −8.39480417701599606346385129512, −7.79854337306656993186043288413, −6.21229037201043475270076670267, −5.97737690296299458680174685045, −5.006490414737415847493299174651, −2.983792643570901574927212509945, −1.86479214244051369644887187624, −1.046797893936335438250812859649, 0.83769814314544155061383436617, 1.97899223323403466795455662340, 3.33755874240185437977830963026, 4.78044350119525238501203753166, 5.71213485844162385008340871084, 6.82407371929972757800346514629, 7.228503964801459106050167399190, 8.70315031051249981298687110028, 9.71081544798197849660469185263, 10.331784282322417956461891394594, 10.9311709943538584984354824689, 11.72678005079091513120419153177, 12.981103895389424854693578031377, 14.04129091236408505495234185496, 15.06459289419372252160268183807, 16.09953692419794620659391581757, 16.586982824257810889096924975094, 17.61803755359629586028415346363, 18.05323893162973818927475879569, 18.59569561768644502306713259230, 20.18085000785180241473342157895, 20.74318995799991477603820954579, 21.32415656099398634259867926531, 22.53183376455924634619410149489, 23.20473255300451491131874589557

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