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Properties

Label 1-569-569.47-r1-0-0
Degree $1$
Conductor $569$
Sign $-0.188 - 0.982i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (0.980 + 0.197i)2-s + (0.165 + 0.986i)3-s + (0.921 + 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.0331 + 0.999i)6-s + (−0.650 − 0.759i)7-s + (0.826 + 0.562i)8-s + (−0.945 + 0.325i)9-s + (0.219 + 0.975i)10-s + (−0.515 − 0.856i)11-s + (−0.230 + 0.973i)12-s + (−0.999 + 0.0221i)13-s + (−0.487 − 0.873i)14-s + (−0.833 + 0.553i)15-s + (0.699 + 0.714i)16-s + (−0.262 − 0.964i)17-s + ⋯
L(s)  = 1  + (0.980 + 0.197i)2-s + (0.165 + 0.986i)3-s + (0.921 + 0.387i)4-s + (0.408 + 0.912i)5-s + (−0.0331 + 0.999i)6-s + (−0.650 − 0.759i)7-s + (0.826 + 0.562i)8-s + (−0.945 + 0.325i)9-s + (0.219 + 0.975i)10-s + (−0.515 − 0.856i)11-s + (−0.230 + 0.973i)12-s + (−0.999 + 0.0221i)13-s + (−0.487 − 0.873i)14-s + (−0.833 + 0.553i)15-s + (0.699 + 0.714i)16-s + (−0.262 − 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.188 - 0.982i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ -0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05081372280 - 0.06146972187i\)
\(L(\frac12)\) \(\approx\) \(0.05081372280 - 0.06146972187i\)
\(L(1)\) \(\approx\) \(1.290459700 + 0.6370346172i\)
\(L(1)\) \(\approx\) \(1.290459700 + 0.6370346172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.980 + 0.197i)T \)
3 \( 1 + (0.165 + 0.986i)T \)
5 \( 1 + (0.408 + 0.912i)T \)
7 \( 1 + (-0.650 - 0.759i)T \)
11 \( 1 + (-0.515 - 0.856i)T \)
13 \( 1 + (-0.999 + 0.0221i)T \)
17 \( 1 + (-0.262 - 0.964i)T \)
19 \( 1 + (-0.925 - 0.377i)T \)
23 \( 1 + (0.766 - 0.641i)T \)
29 \( 1 + (-0.867 + 0.496i)T \)
31 \( 1 + (-0.397 - 0.917i)T \)
37 \( 1 + (0.0773 - 0.997i)T \)
41 \( 1 + (-0.996 - 0.0883i)T \)
43 \( 1 + (0.197 + 0.980i)T \)
47 \( 1 + (-0.794 + 0.607i)T \)
53 \( 1 + (0.999 + 0.0331i)T \)
59 \( 1 + (-0.607 - 0.794i)T \)
61 \( 1 + (-0.304 + 0.952i)T \)
67 \( 1 + (-0.262 + 0.964i)T \)
71 \( 1 + (-0.826 + 0.562i)T \)
73 \( 1 + (-0.377 + 0.925i)T \)
79 \( 1 + (0.997 - 0.0663i)T \)
83 \( 1 + (-0.589 - 0.807i)T \)
89 \( 1 + (0.917 + 0.397i)T \)
97 \( 1 + (0.0552 + 0.998i)T \)
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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

−23.43640744907792967790298696078, −22.51231245734988602896888528994, −21.62965859774140542674530229824, −20.86656018215698369158505654485, −19.89183333653967248473612543786, −19.43014077919351322549227507451, −18.463293848557612592459035049136, −17.25643868316007625819747238538, −16.679441131214440695715565369441, −15.24164613141754956775588375876, −14.93264225879521546009590773534, −13.55559934795512427391277617728, −12.9987688910989033164338220880, −12.4048097981027766283951451676, −11.91004093208085611453448550448, −10.45292822665674212261311193319, −9.46141542020275279320483273968, −8.41108624803304590147426003361, −7.313607348580764045692119102271, −6.39756852974805344522012978035, −5.55727673250490871657669471385, −4.77666519447649744675461023824, −3.36436654166384698654596434435, −2.18404425344410117575596344248, −1.71281384178675816691607962413, 0.0111286355269346989844813886, 2.461525216004640783517240407788, 3.00131131463518531573505394737, 3.95993848504278464679591913166, 4.942649025696713178485764313090, 5.88386419847173100648311216440, 6.8432908549220132537586214491, 7.650228292887346452143072310789, 9.091407495885767189361907644991, 10.13337504115629696012728018370, 10.83921625195044607343880245283, 11.42882982022604607984032675037, 12.954841972306460360379208861079, 13.57671106969777098431875024428, 14.51626613862110933490962727186, 14.94794626347975340186471112990, 16.02810009876446998347747013168, 16.63932329407007810122367515048, 17.41352564958640165776996535352, 18.89113303379514622246212167685, 19.73066431180393791592947844657, 20.59241885532104382715677425256, 21.390326600897657538710923118586, 22.07352753681938056451554094954, 22.6450322482119003985796648634

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