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Properties

Label 1-569-569.172-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.868 + 0.494i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.868 + 0.494i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.868 + 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04092795515 - 0.1545646020i\)
\(L(\frac12)\) \(\approx\) \(-0.04092795515 - 0.1545646020i\)
\(L(1)\) \(\approx\) \(0.4579039114 - 0.1630982959i\)
\(L(1)\) \(\approx\) \(0.4579039114 - 0.1630982959i\)

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

−23.8692297722806255032547961942, −22.22649697441426493687941681227, −21.976077518517122859229717872151, −21.57128552910402304546308656967, −20.369429206967833373085715762891, −19.5502334827812047766916025668, −18.362595365577415326020753090408, −18.17853220832463492903345182209, −17.11651618830057416489893452355, −16.36820405863979485097877258455, −15.42937730609556349346057427646, −14.76553274252391964592324316761, −13.58475686244269380041597375847, −12.07534634182917851558807645920, −11.34328586464755668296203953026, −11.142680744038555407867094761996, −9.86072286506073090629324067209, −9.30984361152746900715988524220, −8.39740109770787560454357157595, −7.05502768448456783592524733063, −6.30928618873163306159343944536, −5.163386607705283935541291666145, −3.840830541301083080537769270214, −2.88637087495061141298138659943, −1.8640450595580695387630506268, 0.1163558348747821365672598064, 1.43962356198486235462368117719, 2.01629479899209757225662346608, 4.29889546922069258691102388627, 5.26072999149627313595223419673, 6.086010899490830630132219210508, 7.312236382082142533428026124377, 7.686010522536615307111595253948, 8.69161851990388684000720332590, 9.82482266149201348514798165930, 10.55934340946368872274084065068, 11.65027606785815813025943129242, 12.403280563042672849428376042364, 13.38724131908432657054922585527, 14.33943295267316466676486474437, 15.34161488270197256143435045691, 16.67212179227906166778650071904, 16.81408267484611784202059567407, 17.86463558527694825211505418260, 18.06015375098069138742022282551, 19.612961350129095677001979637239, 19.99149562889107604523758296948, 20.66539299799582401099552310107, 22.184569487231198259449199076640, 23.09048532948489947124932746480

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