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Properties

Label 1-569-569.107-r0-0-0
Degree $1$
Conductor $569$
Sign $0.0251 + 0.999i$
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  + (0.862 + 0.506i)2-s + (0.903 − 0.428i)3-s + (0.487 + 0.873i)4-s + (0.562 + 0.826i)5-s + (0.996 + 0.0883i)6-s + (−0.666 + 0.745i)7-s + (−0.0221 + 0.999i)8-s + (0.633 − 0.773i)9-s + (0.0663 + 0.997i)10-s + (−0.921 + 0.387i)11-s + (0.814 + 0.580i)12-s + (−0.448 + 0.894i)13-s + (−0.952 + 0.304i)14-s + (0.862 + 0.506i)15-s + (−0.525 + 0.850i)16-s + (0.759 − 0.650i)17-s + ⋯
L(s)  = 1  + (0.862 + 0.506i)2-s + (0.903 − 0.428i)3-s + (0.487 + 0.873i)4-s + (0.562 + 0.826i)5-s + (0.996 + 0.0883i)6-s + (−0.666 + 0.745i)7-s + (−0.0221 + 0.999i)8-s + (0.633 − 0.773i)9-s + (0.0663 + 0.997i)10-s + (−0.921 + 0.387i)11-s + (0.814 + 0.580i)12-s + (−0.448 + 0.894i)13-s + (−0.952 + 0.304i)14-s + (0.862 + 0.506i)15-s + (−0.525 + 0.850i)16-s + (0.759 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0251 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0251 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.0251 + 0.999i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.0251 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.130441570 + 2.184701372i\)
\(L(\frac12)\) \(\approx\) \(2.130441570 + 2.184701372i\)
\(L(1)\) \(\approx\) \(1.942171343 + 1.014481101i\)
\(L(1)\) \(\approx\) \(1.942171343 + 1.014481101i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.862 + 0.506i)T \)
3 \( 1 + (0.903 - 0.428i)T \)
5 \( 1 + (0.562 + 0.826i)T \)
7 \( 1 + (-0.666 + 0.745i)T \)
11 \( 1 + (-0.921 + 0.387i)T \)
13 \( 1 + (-0.448 + 0.894i)T \)
17 \( 1 + (0.759 - 0.650i)T \)
19 \( 1 + (0.487 - 0.873i)T \)
23 \( 1 + (-0.283 - 0.958i)T \)
29 \( 1 + (0.759 + 0.650i)T \)
31 \( 1 + (-0.999 + 0.0442i)T \)
37 \( 1 + (-0.666 + 0.745i)T \)
41 \( 1 + (-0.283 - 0.958i)T \)
43 \( 1 + (0.862 - 0.506i)T \)
47 \( 1 + (0.937 + 0.346i)T \)
53 \( 1 + (0.996 + 0.0883i)T \)
59 \( 1 + (0.937 + 0.346i)T \)
61 \( 1 + (-0.975 + 0.219i)T \)
67 \( 1 + (0.759 + 0.650i)T \)
71 \( 1 + (-0.0221 - 0.999i)T \)
73 \( 1 + (0.487 - 0.873i)T \)
79 \( 1 + (0.984 - 0.176i)T \)
83 \( 1 + (-0.110 - 0.993i)T \)
89 \( 1 + (-0.999 - 0.0442i)T \)
97 \( 1 + (-0.367 + 0.930i)T \)
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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

−23.02035318962374935985878116739, −21.96956545625830582811691417919, −21.23519267709695238894788081886, −20.69106481764509623606143840515, −19.89960338752260966355383414568, −19.40464355061648943946050433441, −18.26578591010740183728122393893, −16.87139631462790934755456962504, −16.07761762849234301373719541169, −15.41264567329894640392962138089, −14.26251689375542422940608873643, −13.6876854736324257715619866689, −12.92668932729017657376826911600, −12.398004773057206450709940377972, −10.79262909569040299976329304270, −10.01230862048843995492820114646, −9.65345984829970679203819665589, −8.223569543886583672940984999, −7.38657553338428664094458971939, −5.81279405454378778221924695918, −5.242963980387169710654747845432, −4.02702435557557330859815018944, −3.322538038171682254411265259225, −2.307939780494263752117124130297, −1.085289906756322619732027647345, 2.17910005001924256715661811831, 2.657020363711479570535056057962, 3.49290639813164620255019971096, 4.90204950849012286535276583712, 5.906774028025647642382174111427, 7.00934195649182306445515528516, 7.26282561712186852827951262254, 8.63716916047017953805341551019, 9.47127554368076881997885422905, 10.53391034855273848958947264514, 11.98333025164083667501778291150, 12.535734051686450115839422742580, 13.575945014682626274718711464915, 14.06834212230295297951942998876, 14.91776966959863650620862420243, 15.59020789337402084052044239426, 16.45249970638957465088413150298, 17.79918868724612287916160767187, 18.45355502343853052582764183340, 19.22992701797128009555445082504, 20.373608109949598350435026768681, 21.12437676358398841003698699956, 21.91174086924703007468055015111, 22.54231053024340436610332537707, 23.6003591960000262048964053509

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