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Properties

Label 1-569-569.177-r0-0-0
Degree $1$
Conductor $569$
Sign $0.882 - 0.470i$
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  + (−0.903 + 0.428i)2-s + (0.988 − 0.154i)3-s + (0.633 − 0.773i)4-s + (−0.283 − 0.958i)5-s + (−0.826 + 0.562i)6-s + (−0.984 − 0.176i)7-s + (−0.240 + 0.970i)8-s + (0.952 − 0.304i)9-s + (0.666 + 0.745i)10-s + (0.945 + 0.325i)11-s + (0.506 − 0.862i)12-s + (0.921 + 0.387i)13-s + (0.964 − 0.262i)14-s + (−0.428 − 0.903i)15-s + (−0.197 − 0.980i)16-s + (−0.0663 + 0.997i)17-s + ⋯
L(s)  = 1  + (−0.903 + 0.428i)2-s + (0.988 − 0.154i)3-s + (0.633 − 0.773i)4-s + (−0.283 − 0.958i)5-s + (−0.826 + 0.562i)6-s + (−0.984 − 0.176i)7-s + (−0.240 + 0.970i)8-s + (0.952 − 0.304i)9-s + (0.666 + 0.745i)10-s + (0.945 + 0.325i)11-s + (0.506 − 0.862i)12-s + (0.921 + 0.387i)13-s + (0.964 − 0.262i)14-s + (−0.428 − 0.903i)15-s + (−0.197 − 0.980i)16-s + (−0.0663 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224797231 - 0.3062769699i\)
\(L(\frac12)\) \(\approx\) \(1.224797231 - 0.3062769699i\)
\(L(1)\) \(\approx\) \(0.9883198779 - 0.08206185087i\)
\(L(1)\) \(\approx\) \(0.9883198779 - 0.08206185087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.903 + 0.428i)T \)
3 \( 1 + (0.988 - 0.154i)T \)
5 \( 1 + (-0.283 - 0.958i)T \)
7 \( 1 + (-0.984 - 0.176i)T \)
11 \( 1 + (0.945 + 0.325i)T \)
13 \( 1 + (0.921 + 0.387i)T \)
17 \( 1 + (-0.0663 + 0.997i)T \)
19 \( 1 + (0.773 - 0.633i)T \)
23 \( 1 + (0.999 - 0.0221i)T \)
29 \( 1 + (-0.997 + 0.0663i)T \)
31 \( 1 + (-0.467 - 0.883i)T \)
37 \( 1 + (-0.176 + 0.984i)T \)
41 \( 1 + (-0.0221 - 0.999i)T \)
43 \( 1 + (0.903 + 0.428i)T \)
47 \( 1 + (0.683 - 0.730i)T \)
53 \( 1 + (0.826 - 0.562i)T \)
59 \( 1 + (-0.683 + 0.730i)T \)
61 \( 1 + (-0.759 - 0.650i)T \)
67 \( 1 + (-0.0663 - 0.997i)T \)
71 \( 1 + (-0.240 - 0.970i)T \)
73 \( 1 + (-0.773 + 0.633i)T \)
79 \( 1 + (0.367 + 0.930i)T \)
83 \( 1 + (-0.346 + 0.937i)T \)
89 \( 1 + (0.467 - 0.883i)T \)
97 \( 1 + (-0.544 - 0.839i)T \)
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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

−23.05408366619180277141237633759, −22.3164953256604790099664738828, −21.55107845556354839213297563013, −20.49983051191001932752653184804, −19.92491069504577167694407526247, −18.98369326641192951489200064935, −18.73467479085347521764418885903, −17.795664544172421000930646425043, −16.36694021932664818452542428258, −15.93124217440437741275893002869, −15.00558868728739249516175614299, −14.013865840154877317737950089108, −13.11810761243582009683613307382, −12.07656128341515927313486687725, −11.07535587090567264519679073958, −10.30492688347874340888892425386, −9.29067229209245738648933536322, −8.92376087865642740025980646775, −7.5963342286924283014799348434, −7.06448460076641562260907654440, −6.02244647182329239521503008721, −3.86719505450891381761288679360, −3.31025935336613310061773314359, −2.61956030321090063446024970584, −1.20629886109438099734518969753, 0.964144934465337804497075653062, 1.864161961591617711947932974077, 3.3575637517503844736870605912, 4.27575998900425180149021170630, 5.792753313015077621861696174051, 6.81206590993720563600947900436, 7.54093166287924056509875934016, 8.68901180574923835252947157493, 9.11100187227568350780666084877, 9.7528925663536969859969090553, 11.00547706848764423614567634650, 12.15279895646627907149582560264, 13.11745018094775228178213344826, 13.86599673319346905912963140856, 15.117498049910268049771447984814, 15.57248894942076444973246215978, 16.59837322273111142357161115250, 17.08516733870407183695089142791, 18.37799184583551533738152292884, 19.170920601995400970899595640011, 19.75108021735194627421604640678, 20.34792891211673640071579457928, 21.1114945564251908892490349057, 22.47372697126009039908228888415, 23.61064071517894369639670893852

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