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Properties

Label 1-569-569.43-r0-0-0
Degree $1$
Conductor $569$
Sign $0.337 + 0.941i$
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  + (−0.883 − 0.467i)2-s + (0.598 − 0.801i)3-s + (0.562 + 0.826i)4-s + (0.154 + 0.988i)5-s + (−0.903 + 0.428i)6-s + (0.487 + 0.873i)7-s + (−0.110 − 0.993i)8-s + (−0.283 − 0.958i)9-s + (0.325 − 0.945i)10-s + (−0.408 + 0.912i)11-s + (0.999 + 0.0442i)12-s + (−0.730 + 0.683i)13-s + (−0.0221 − 0.999i)14-s + (0.883 + 0.467i)15-s + (−0.367 + 0.930i)16-s + (−0.921 − 0.387i)17-s + ⋯
L(s)  = 1  + (−0.883 − 0.467i)2-s + (0.598 − 0.801i)3-s + (0.562 + 0.826i)4-s + (0.154 + 0.988i)5-s + (−0.903 + 0.428i)6-s + (0.487 + 0.873i)7-s + (−0.110 − 0.993i)8-s + (−0.283 − 0.958i)9-s + (0.325 − 0.945i)10-s + (−0.408 + 0.912i)11-s + (0.999 + 0.0442i)12-s + (−0.730 + 0.683i)13-s + (−0.0221 − 0.999i)14-s + (0.883 + 0.467i)15-s + (−0.367 + 0.930i)16-s + (−0.921 − 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.337 + 0.941i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6894090409 + 0.4849362977i\)
\(L(\frac12)\) \(\approx\) \(0.6894090409 + 0.4849362977i\)
\(L(1)\) \(\approx\) \(0.7982435985 + 0.02878541169i\)
\(L(1)\) \(\approx\) \(0.7982435985 + 0.02878541169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.883 - 0.467i)T \)
3 \( 1 + (0.598 - 0.801i)T \)
5 \( 1 + (0.154 + 0.988i)T \)
7 \( 1 + (0.487 + 0.873i)T \)
11 \( 1 + (-0.408 + 0.912i)T \)
13 \( 1 + (-0.730 + 0.683i)T \)
17 \( 1 + (-0.921 - 0.387i)T \)
19 \( 1 + (-0.562 + 0.826i)T \)
23 \( 1 + (0.991 - 0.132i)T \)
29 \( 1 + (0.921 - 0.387i)T \)
31 \( 1 + (0.975 + 0.219i)T \)
37 \( 1 + (-0.487 - 0.873i)T \)
41 \( 1 + (-0.991 + 0.132i)T \)
43 \( 1 + (-0.883 + 0.467i)T \)
47 \( 1 + (0.197 + 0.980i)T \)
53 \( 1 + (-0.903 + 0.428i)T \)
59 \( 1 + (0.197 + 0.980i)T \)
61 \( 1 + (-0.448 - 0.894i)T \)
67 \( 1 + (-0.921 + 0.387i)T \)
71 \( 1 + (-0.110 + 0.993i)T \)
73 \( 1 + (-0.562 + 0.826i)T \)
79 \( 1 + (0.633 + 0.773i)T \)
83 \( 1 + (0.525 - 0.850i)T \)
89 \( 1 + (0.975 - 0.219i)T \)
97 \( 1 + (0.952 - 0.304i)T \)
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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

−23.49536247122264549231959942255, −22.063052167626830890888873662438, −21.11413834665142402214521498392, −20.48414626645460487547553807351, −19.7503095155210423404922521946, −19.2196945137368867202507357450, −17.73093008115160491021921859966, −17.09097261673222307178657350220, −16.54421931469864231548588302884, −15.51898302103871077722619407108, −15.03633051397867858042222177571, −13.78312407633270856192875459712, −13.28298507821161693331246331466, −11.631655463937546220143949313166, −10.613788938520076059041152713049, −10.167553704420826887953942750343, −8.96302455859986197337013193549, −8.47651455388229899119843288563, −7.74671843960720455091358148575, −6.519583756567832294214811210396, −5.044631541903688385766706902317, −4.76441507608002099756719369738, −3.15222087728223614508173438690, −1.898466250988457154318682068497, −0.506728974046794909569601476586, 1.69325532024543263560802542826, 2.35656907861453036396419185656, 3.00118969197940776056955069322, 4.53441324069027844358217939253, 6.298906118934433825056254634889, 6.9851829658807055794800259753, 7.77835115643318535531970873313, 8.68427659347227497819217472482, 9.5133755523300222371273111535, 10.41566753366802579340791089645, 11.53791539131574302221248680559, 12.13547690467388026040181571253, 13.01385155995631970013805848765, 14.18252904018787602328475826975, 14.99954425187584271485522015759, 15.67291145959008626036836940189, 17.30274749642956804941259844430, 17.74533075886153833910408417193, 18.59875033027763751040855930038, 19.02380795187330588525929036947, 19.83343044641027766653039772523, 20.89079269220474144367289593971, 21.43743525489646389444001822443, 22.48895358133066913497414777402, 23.45192910899142499889716145026

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