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Properties

Label 1-569-569.68-r0-0-0
Degree $1$
Conductor $569$
Sign $0.990 - 0.137i$
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  + (0.197 − 0.980i)2-s + (0.580 − 0.814i)3-s + (−0.921 − 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.683 − 0.730i)6-s + (−0.759 + 0.650i)7-s + (−0.562 + 0.826i)8-s + (−0.325 − 0.945i)9-s + (0.975 − 0.219i)10-s + (0.970 + 0.240i)11-s + (−0.850 + 0.525i)12-s + (0.0221 + 0.999i)13-s + (0.487 + 0.873i)14-s + (0.980 + 0.197i)15-s + (0.699 + 0.714i)16-s + (−0.964 + 0.262i)17-s + ⋯
L(s)  = 1  + (0.197 − 0.980i)2-s + (0.580 − 0.814i)3-s + (−0.921 − 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.683 − 0.730i)6-s + (−0.759 + 0.650i)7-s + (−0.562 + 0.826i)8-s + (−0.325 − 0.945i)9-s + (0.975 − 0.219i)10-s + (0.970 + 0.240i)11-s + (−0.850 + 0.525i)12-s + (0.0221 + 0.999i)13-s + (0.487 + 0.873i)14-s + (0.980 + 0.197i)15-s + (0.699 + 0.714i)16-s + (−0.964 + 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.990 - 0.137i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.990 - 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421001436 - 0.09807091101i\)
\(L(\frac12)\) \(\approx\) \(1.421001436 - 0.09807091101i\)
\(L(1)\) \(\approx\) \(1.128168311 - 0.4028245796i\)
\(L(1)\) \(\approx\) \(1.128168311 - 0.4028245796i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.197 - 0.980i)T \)
3 \( 1 + (0.580 - 0.814i)T \)
5 \( 1 + (0.408 + 0.912i)T \)
7 \( 1 + (-0.759 + 0.650i)T \)
11 \( 1 + (0.970 + 0.240i)T \)
13 \( 1 + (0.0221 + 0.999i)T \)
17 \( 1 + (-0.964 + 0.262i)T \)
19 \( 1 + (0.387 + 0.921i)T \)
23 \( 1 + (-0.0883 + 0.996i)T \)
29 \( 1 + (-0.262 + 0.964i)T \)
31 \( 1 + (-0.930 - 0.367i)T \)
37 \( 1 + (0.650 + 0.759i)T \)
41 \( 1 + (0.996 + 0.0883i)T \)
43 \( 1 + (-0.197 - 0.980i)T \)
47 \( 1 + (0.132 - 0.991i)T \)
53 \( 1 + (0.683 + 0.730i)T \)
59 \( 1 + (-0.132 + 0.991i)T \)
61 \( 1 + (0.952 + 0.304i)T \)
67 \( 1 + (-0.964 - 0.262i)T \)
71 \( 1 + (-0.562 - 0.826i)T \)
73 \( 1 + (-0.387 - 0.921i)T \)
79 \( 1 + (-0.0663 - 0.997i)T \)
83 \( 1 + (0.988 + 0.154i)T \)
89 \( 1 + (0.930 - 0.367i)T \)
97 \( 1 + (-0.745 - 0.666i)T \)
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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

−23.22598392621906117339871709651, −22.32311247192098910554707468059, −21.947337030392209216526211391808, −20.76259990835301424712204240463, −20.02449316402209818439857472150, −19.36801163487836851137845433670, −17.82007315051737198267212456685, −17.18904917807349149153035090463, −16.23802826674947763485728860869, −15.98438920874798622603534089131, −14.86705192350887070599218647314, −14.046281950954426993046506317, −13.24991796913533605569961570701, −12.72707639623577371385536103976, −11.133247975638734705621639833818, −9.87732231854444226540430674321, −9.28696718709018024785724874048, −8.62407704712534193855251486134, −7.62030017328932982480851216045, −6.484573813140287881540025019919, −5.53756711843672923132626680855, −4.50466320728053672046279685503, −3.90705244484876179853076474215, −2.70787530573921405378264860732, −0.649277137741326661701058793754, 1.59896167069711397739125880848, 2.19377463076325386510701486537, 3.27365153232491062018733482263, 3.95748693207942564099503646322, 5.74691077771743339332240289940, 6.450693061992390424603403266883, 7.404112986662085310281833953452, 9.03104986562644835884805536714, 9.202750997157676595297377053793, 10.304168019902588814293153235326, 11.56826284571147167018630007008, 12.01936741495952912336235401744, 13.10822694007104085463334227204, 13.721082235986945920065105038657, 14.57577872954819167595083453785, 15.13702472039146449781584801389, 16.76059369501417772451272611617, 17.93964724784426219966065400706, 18.39919759188426282234073324816, 19.236674024804821559636920002712, 19.66662475856612471310070083786, 20.63210592742802145446852674996, 21.81262388901309777002461686374, 22.1294674453954301061602855628, 23.08202700054470336359228188991

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