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Properties

Label 1-569-569.80-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.396 + 0.918i$
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  + (−0.975 + 0.219i)2-s + (0.759 + 0.650i)3-s + (0.903 − 0.428i)4-s + (−0.598 + 0.801i)5-s + (−0.883 − 0.467i)6-s + (0.996 + 0.0883i)7-s + (−0.787 + 0.616i)8-s + (0.154 + 0.988i)9-s + (0.408 − 0.912i)10-s + (0.814 − 0.580i)11-s + (0.964 + 0.262i)12-s + (−0.197 + 0.980i)13-s + (−0.991 + 0.132i)14-s + (−0.975 + 0.219i)15-s + (0.633 − 0.773i)16-s + (−0.730 + 0.683i)17-s + ⋯
L(s)  = 1  + (−0.975 + 0.219i)2-s + (0.759 + 0.650i)3-s + (0.903 − 0.428i)4-s + (−0.598 + 0.801i)5-s + (−0.883 − 0.467i)6-s + (0.996 + 0.0883i)7-s + (−0.787 + 0.616i)8-s + (0.154 + 0.988i)9-s + (0.408 − 0.912i)10-s + (0.814 − 0.580i)11-s + (0.964 + 0.262i)12-s + (−0.197 + 0.980i)13-s + (−0.991 + 0.132i)14-s + (−0.975 + 0.219i)15-s + (0.633 − 0.773i)16-s + (−0.730 + 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.396 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6579764061 + 1.000854055i\)
\(L(\frac12)\) \(\approx\) \(0.6579764061 + 1.000854055i\)
\(L(1)\) \(\approx\) \(0.8130666539 + 0.4954693539i\)
\(L(1)\) \(\approx\) \(0.8130666539 + 0.4954693539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.975 + 0.219i)T \)
3 \( 1 + (0.759 + 0.650i)T \)
5 \( 1 + (-0.598 + 0.801i)T \)
7 \( 1 + (0.996 + 0.0883i)T \)
11 \( 1 + (0.814 - 0.580i)T \)
13 \( 1 + (-0.197 + 0.980i)T \)
17 \( 1 + (-0.730 + 0.683i)T \)
19 \( 1 + (0.903 + 0.428i)T \)
23 \( 1 + (0.699 - 0.714i)T \)
29 \( 1 + (-0.730 - 0.683i)T \)
31 \( 1 + (0.240 + 0.970i)T \)
37 \( 1 + (0.996 + 0.0883i)T \)
41 \( 1 + (0.699 - 0.714i)T \)
43 \( 1 + (-0.975 - 0.219i)T \)
47 \( 1 + (-0.367 + 0.930i)T \)
53 \( 1 + (-0.883 - 0.467i)T \)
59 \( 1 + (-0.367 + 0.930i)T \)
61 \( 1 + (0.937 + 0.346i)T \)
67 \( 1 + (-0.730 - 0.683i)T \)
71 \( 1 + (-0.787 - 0.616i)T \)
73 \( 1 + (0.903 + 0.428i)T \)
79 \( 1 + (0.562 - 0.826i)T \)
83 \( 1 + (0.984 + 0.176i)T \)
89 \( 1 + (0.240 - 0.970i)T \)
97 \( 1 + (-0.283 - 0.958i)T \)
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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

−23.29413167334037708551962324364, −21.94910414392207475904466080554, −20.7478736489829872331100248204, −20.16415563463507710843247428495, −19.94765090172676089882539553784, −18.87586340312869623657604632029, −17.88209399031358138572731343503, −17.50983819608017004128200175577, −16.4376540194073966494885909721, −15.2710979992407186772696757200, −14.87899016387006696977679973856, −13.45156268416855946689365821601, −12.65380560057717889511214503635, −11.69588179541940345935211449474, −11.23014695843105912738602599021, −9.572590830239172631730248621248, −9.13755552955646074144451214191, −8.07405264645560443996226756165, −7.63661476184993373845946561869, −6.79524499918690851242018730610, −5.20187865872750373745449824645, −3.95167730757007648296672751707, −2.8200398348543245886555557574, −1.63289548546198659522388820989, −0.853380051322389327925293513373, 1.53655352337680725256099774926, 2.558526878447465002164529114866, 3.679037969513904334916530845610, 4.70687682048876531894549680419, 6.17500282487048838402399710159, 7.18588969001267330887372025158, 8.015026752519308370589354945880, 8.745483033741191456974696532444, 9.51359468191111762084807648850, 10.66490980850744228865738842682, 11.194364499762256010127992599669, 11.95144642836248215688637574845, 13.83474904195145326043427250628, 14.61946018466895260891375869704, 14.966496500379128883440819077128, 16.01055134097868941836206181560, 16.70946893344773156117467924669, 17.71763853994594810670960192144, 18.72632182942512014153033247609, 19.26127965585498492094788337473, 19.99664227452117183381855727812, 20.89918479124011782841551500292, 21.65266047247384512476734689042, 22.54511756016098556535514814754, 23.91297743304852888609991632478

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