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Properties

Label 1-572-572.35-r0-0-0
Degree $1$
Conductor $572$
Sign $0.651 + 0.758i$
Analytic cond. $2.65635$
Root an. cond. $2.65635$
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  + (0.978 − 0.207i)3-s + (−0.809 + 0.587i)5-s + (−0.978 − 0.207i)7-s + (0.913 − 0.406i)9-s + (−0.669 + 0.743i)15-s + (0.104 + 0.994i)17-s + (0.669 + 0.743i)19-s − 21-s + (0.5 + 0.866i)23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−0.669 + 0.743i)29-s + (0.809 + 0.587i)31-s + (0.913 − 0.406i)35-s + (0.669 − 0.743i)37-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)3-s + (−0.809 + 0.587i)5-s + (−0.978 − 0.207i)7-s + (0.913 − 0.406i)9-s + (−0.669 + 0.743i)15-s + (0.104 + 0.994i)17-s + (0.669 + 0.743i)19-s − 21-s + (0.5 + 0.866i)23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−0.669 + 0.743i)29-s + (0.809 + 0.587i)31-s + (0.913 − 0.406i)35-s + (0.669 − 0.743i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.651 + 0.758i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ 0.651 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.334757853 + 0.6135382004i\)
\(L(\frac12)\) \(\approx\) \(1.334757853 + 0.6135382004i\)
\(L(1)\) \(\approx\) \(1.182533990 + 0.1663491076i\)
\(L(1)\) \(\approx\) \(1.182533990 + 0.1663491076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−23.01393147566546022464308213808, −22.37823064450474883372761934653, −21.31890713576871468857496281143, −20.321606136791560899888166435182, −20.03699204327356066297154254833, −18.97272152068197559578101863338, −18.59075435364130574494716022856, −17.01840128454708535693646450748, −16.13681740857345613137320267644, −15.61589533522723808507193714843, −14.85469015429367721375712544067, −13.65227527545080225124545983116, −13.066418227998894734878032261555, −12.1517086684674163238152070421, −11.19465940117787179866585386955, −9.82969244059579033822568784000, −9.301931543774792383555716073844, −8.412542195868847228046968129818, −7.53063961997918966898666349706, −6.660216752386687207991532979387, −5.15413803443035072362564347134, −4.246766013079836886809042606662, −3.266406012514267161154939051335, −2.478902800918758514901050756350, −0.7486259740067949637801691918, 1.32152476670773215641017651042, 2.81225182940334130999974150963, 3.46180256275446793497933827368, 4.22525531183543741630907443193, 5.91795451472039056077403224638, 6.95976158795289619834883859248, 7.604525952767788784282178581873, 8.49871441058074998213537820264, 9.55216109342539233205848978470, 10.29061725975540379928402891820, 11.37952973613179269893280446851, 12.530837269070223303910306913833, 13.0792115390726152135868387243, 14.224277067389605916315422582527, 14.80398395822898244719889924686, 15.76307004247503882233662281286, 16.31265361964034280574166767545, 17.692853720155053802114849649129, 18.65954116083487464923318114026, 19.392756921536142607890170829824, 19.70741615901187887128504290637, 20.72931789463417338353965938776, 21.68137410541127127517028686852, 22.62320719222989365118112687521, 23.36198944385477102006676532077

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