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Properties

Label 1-569-569.16-r0-0-0
Degree $1$
Conductor $569$
Sign $0.934 + 0.357i$
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  + (−0.367 − 0.930i)2-s + (−0.839 − 0.544i)3-s + (−0.730 + 0.683i)4-s + (0.814 + 0.580i)5-s + (−0.197 + 0.980i)6-s + (−0.448 + 0.894i)7-s + (0.903 + 0.428i)8-s + (0.408 + 0.912i)9-s + (0.240 − 0.970i)10-s + (−0.110 − 0.993i)11-s + (0.984 − 0.176i)12-s + (−0.991 + 0.132i)13-s + (0.996 + 0.0883i)14-s + (−0.367 − 0.930i)15-s + (0.0663 − 0.997i)16-s + (−0.0221 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.367 − 0.930i)2-s + (−0.839 − 0.544i)3-s + (−0.730 + 0.683i)4-s + (0.814 + 0.580i)5-s + (−0.197 + 0.980i)6-s + (−0.448 + 0.894i)7-s + (0.903 + 0.428i)8-s + (0.408 + 0.912i)9-s + (0.240 − 0.970i)10-s + (−0.110 − 0.993i)11-s + (0.984 − 0.176i)12-s + (−0.991 + 0.132i)13-s + (0.996 + 0.0883i)14-s + (−0.367 − 0.930i)15-s + (0.0663 − 0.997i)16-s + (−0.0221 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.934 + 0.357i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.934 + 0.357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6082416735 + 0.1123019246i\)
\(L(\frac12)\) \(\approx\) \(0.6082416735 + 0.1123019246i\)
\(L(1)\) \(\approx\) \(0.6182388376 - 0.1653782227i\)
\(L(1)\) \(\approx\) \(0.6182388376 - 0.1653782227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.367 - 0.930i)T \)
3 \( 1 + (-0.839 - 0.544i)T \)
5 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (-0.448 + 0.894i)T \)
11 \( 1 + (-0.110 - 0.993i)T \)
13 \( 1 + (-0.991 + 0.132i)T \)
17 \( 1 + (-0.0221 - 0.999i)T \)
19 \( 1 + (-0.730 - 0.683i)T \)
23 \( 1 + (0.862 + 0.506i)T \)
29 \( 1 + (-0.0221 + 0.999i)T \)
31 \( 1 + (0.633 - 0.773i)T \)
37 \( 1 + (-0.448 + 0.894i)T \)
41 \( 1 + (0.862 + 0.506i)T \)
43 \( 1 + (-0.367 + 0.930i)T \)
47 \( 1 + (0.699 + 0.714i)T \)
53 \( 1 + (-0.197 + 0.980i)T \)
59 \( 1 + (0.699 + 0.714i)T \)
61 \( 1 + (-0.283 + 0.958i)T \)
67 \( 1 + (-0.0221 + 0.999i)T \)
71 \( 1 + (0.903 - 0.428i)T \)
73 \( 1 + (-0.730 - 0.683i)T \)
79 \( 1 + (-0.921 + 0.387i)T \)
83 \( 1 + (-0.598 - 0.801i)T \)
89 \( 1 + (0.633 + 0.773i)T \)
97 \( 1 + (0.325 + 0.945i)T \)
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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

−23.13051439942198004369760335278, −22.72954008255552934855044765333, −21.68961023451623898799786020157, −20.82501646834448434913547651439, −19.84494567463151765525799582220, −18.851247773385290979326337787907, −17.50796090834397883527964615723, −17.311952095827559072255751921053, −16.76849946754726699173277464437, −15.81721990293377093291965257203, −14.93927401781006672537568632357, −14.1238796563676395451189027147, −12.86181281185815693692486048154, −12.47727830042522332051746069758, −10.661465970581350033454141140358, −10.11784402008753929123063593683, −9.57747396768866701738571918491, −8.47600961790123726201252478413, −7.18650131017214913380711526394, −6.502026783991287342881991908759, −5.56552804802240739884690288718, −4.716278285613584741828329845702, −4.00588260481125767610631228749, −1.84450526099727464572069571656, −0.46385669627775140613259727329, 1.1384176478044545122120621905, 2.507662859569275540016382838029, 2.88354730527727927593277344701, 4.74979312748578549204337079805, 5.57458958069716781350913686962, 6.5876164291967861772115899976, 7.52671139128709178165403248132, 8.86459404377705178570053115483, 9.59688332819253771939891630996, 10.56235807647409487539977502192, 11.33473765771967298055523467045, 12.01614966749713300317480557411, 13.0500376168492653277864297107, 13.50821879958283773636433720084, 14.65850323753363462611673161224, 16.04218350990817922317098166827, 16.98198098377961524258953764288, 17.60282733316671063417884808454, 18.50370220480310819872901305915, 18.92635522668944100660645329797, 19.64544412802213651093920474700, 21.14285912687565877589200014701, 21.71010908234328973510628040398, 22.270141008167203911511781539845, 22.91384437027037629702433312496

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