Properties

Label 1-569-569.33-r0-0-0
Degree $1$
Conductor $569$
Sign $0.742 + 0.669i$
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  + (0.487 − 0.873i)2-s + (0.633 + 0.773i)3-s + (−0.525 − 0.850i)4-s + (−0.367 − 0.930i)5-s + (0.984 − 0.176i)6-s + (−0.110 + 0.993i)7-s + (−0.999 + 0.0442i)8-s + (−0.197 + 0.980i)9-s + (−0.991 − 0.132i)10-s + (0.699 + 0.714i)11-s + (0.325 − 0.945i)12-s + (−0.598 + 0.801i)13-s + (0.814 + 0.580i)14-s + (0.487 − 0.873i)15-s + (−0.448 + 0.894i)16-s + (0.154 + 0.988i)17-s + ⋯
L(s)  = 1  + (0.487 − 0.873i)2-s + (0.633 + 0.773i)3-s + (−0.525 − 0.850i)4-s + (−0.367 − 0.930i)5-s + (0.984 − 0.176i)6-s + (−0.110 + 0.993i)7-s + (−0.999 + 0.0442i)8-s + (−0.197 + 0.980i)9-s + (−0.991 − 0.132i)10-s + (0.699 + 0.714i)11-s + (0.325 − 0.945i)12-s + (−0.598 + 0.801i)13-s + (0.814 + 0.580i)14-s + (0.487 − 0.873i)15-s + (−0.448 + 0.894i)16-s + (0.154 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.742 + 0.669i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.742 + 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.415330866 + 0.5441150808i\)
\(L(\frac12)\) \(\approx\) \(1.415330866 + 0.5441150808i\)
\(L(1)\) \(\approx\) \(1.295144740 - 0.06926050262i\)
\(L(1)\) \(\approx\) \(1.295144740 - 0.06926050262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.487 - 0.873i)T \)
3 \( 1 + (0.633 + 0.773i)T \)
5 \( 1 + (-0.367 - 0.930i)T \)
7 \( 1 + (-0.110 + 0.993i)T \)
11 \( 1 + (0.699 + 0.714i)T \)
13 \( 1 + (-0.598 + 0.801i)T \)
17 \( 1 + (0.154 + 0.988i)T \)
19 \( 1 + (-0.525 + 0.850i)T \)
23 \( 1 + (-0.839 - 0.544i)T \)
29 \( 1 + (0.154 - 0.988i)T \)
31 \( 1 + (0.996 + 0.0883i)T \)
37 \( 1 + (-0.110 + 0.993i)T \)
41 \( 1 + (-0.839 - 0.544i)T \)
43 \( 1 + (0.487 + 0.873i)T \)
47 \( 1 + (0.759 - 0.650i)T \)
53 \( 1 + (0.984 - 0.176i)T \)
59 \( 1 + (0.759 - 0.650i)T \)
61 \( 1 + (0.903 + 0.428i)T \)
67 \( 1 + (0.154 - 0.988i)T \)
71 \( 1 + (-0.999 - 0.0442i)T \)
73 \( 1 + (-0.525 + 0.850i)T \)
79 \( 1 + (0.937 + 0.346i)T \)
83 \( 1 + (-0.975 - 0.219i)T \)
89 \( 1 + (0.996 - 0.0883i)T \)
97 \( 1 + (-0.730 + 0.683i)T \)
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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

−23.39744540959036758255808271847, −22.50631071499833753356764290716, −21.85126343155726995401687041880, −20.57924835840201382233181497364, −19.70810989854167863489694539560, −19.03729291215332117762318474788, −17.9019691026634389723565375771, −17.47049375568624488991744996694, −16.30648919002999106987933586047, −15.37438771158279081940416531335, −14.52629740403540802811464631462, −13.94034710719643334293210063814, −13.36003037108400710902974241361, −12.242839414350954686595278728493, −11.424773312710785678765841057007, −10.13054999357688558490035012268, −8.967854419681508720369679238229, −7.98929339837672833600770039952, −7.20066961367079726334490932276, −6.80459512787524849754838078163, −5.7328715068540874359704731394, −4.235351090990439864519206817442, −3.38254310143090681099065979483, −2.632230600460147358501931320753, −0.60713042267936538658439040057, 1.70075014052624189316860941999, 2.415278915669250385505892086029, 3.86033262780642218276354629221, 4.3227696337981094511084309363, 5.244256878907641544185143633196, 6.31537614155898643631593260758, 8.19016850625820944480731045183, 8.76820727836684766811891836272, 9.67930336934299198114166557698, 10.21480571016355686050866451797, 11.7432888737655879335898814301, 12.08935746774148127918423896343, 13.00230844715675074227562918460, 14.09495116556494855053375454198, 14.896602910960844748440115146627, 15.441077018862614350065796176415, 16.53384970923668924605569693074, 17.4298906963585174951500527156, 18.98255861698308713415017332286, 19.29439243265663068965360860773, 20.19033055929889211753038088172, 20.89374712397440513106970652847, 21.56419266819344809985533273228, 22.23769348200890735734502992757, 23.11529989452507196383933889889

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