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Properties

Label 1-571-571.84-r0-0-0
Degree $1$
Conductor $571$
Sign $0.710 - 0.703i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (0.993 + 0.110i)2-s + (−0.421 + 0.906i)3-s + (0.975 + 0.218i)4-s + (−0.0385 − 0.999i)5-s + (−0.518 + 0.854i)6-s + (0.0495 − 0.998i)7-s + (0.945 + 0.324i)8-s + (−0.644 − 0.764i)9-s + (0.0715 − 0.997i)10-s + (−0.899 − 0.436i)11-s + (−0.609 + 0.792i)12-s + (−0.782 + 0.622i)13-s + (0.159 − 0.987i)14-s + (0.922 + 0.386i)15-s + (0.904 + 0.426i)16-s + (0.685 − 0.728i)17-s + ⋯
L(s)  = 1  + (0.993 + 0.110i)2-s + (−0.421 + 0.906i)3-s + (0.975 + 0.218i)4-s + (−0.0385 − 0.999i)5-s + (−0.518 + 0.854i)6-s + (0.0495 − 0.998i)7-s + (0.945 + 0.324i)8-s + (−0.644 − 0.764i)9-s + (0.0715 − 0.997i)10-s + (−0.899 − 0.436i)11-s + (−0.609 + 0.792i)12-s + (−0.782 + 0.622i)13-s + (0.159 − 0.987i)14-s + (0.922 + 0.386i)15-s + (0.904 + 0.426i)16-s + (0.685 − 0.728i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.710 - 0.703i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.710 - 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.826759010 - 0.7511615458i\)
\(L(\frac12)\) \(\approx\) \(1.826759010 - 0.7511615458i\)
\(L(1)\) \(\approx\) \(1.558088400 - 0.1037946923i\)
\(L(1)\) \(\approx\) \(1.558088400 - 0.1037946923i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.993 + 0.110i)T \)
3 \( 1 + (-0.421 + 0.906i)T \)
5 \( 1 + (-0.0385 - 0.999i)T \)
7 \( 1 + (0.0495 - 0.998i)T \)
11 \( 1 + (-0.899 - 0.436i)T \)
13 \( 1 + (-0.782 + 0.622i)T \)
17 \( 1 + (0.685 - 0.728i)T \)
19 \( 1 + (0.874 - 0.485i)T \)
23 \( 1 + (-0.956 - 0.293i)T \)
29 \( 1 + (0.635 - 0.771i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (0.999 + 0.0440i)T \)
41 \( 1 + (-0.962 - 0.272i)T \)
43 \( 1 + (0.970 - 0.240i)T \)
47 \( 1 + (-0.821 + 0.569i)T \)
53 \( 1 + (0.202 - 0.979i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (0.952 - 0.303i)T \)
67 \( 1 + (-0.949 + 0.314i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.537 + 0.843i)T \)
79 \( 1 + (0.802 - 0.596i)T \)
83 \( 1 + (0.999 + 0.0220i)T \)
89 \( 1 + (-0.480 - 0.876i)T \)
97 \( 1 + (0.0935 + 0.995i)T \)
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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

−23.45554435814859726076631680172, −22.38567219188237047025074337226, −22.08097406874647543442882515401, −21.13864538344982886407401298048, −19.89484281293537166671256379118, −19.26899695175585492239896657802, −18.255004083575297773854554558, −17.85717681293779892195842791768, −16.45141643552847249971662904848, −15.53949697902251187371340574418, −14.719836425347438724510817477, −14.08101444196988213594070660784, −12.98922113101287929548781733260, −12.260144446365960932658835944877, −11.76646478860760013205833296678, −10.64513330285034050105743497099, −9.997827436867426082093262114620, −8.01073127006665825068393591965, −7.51143142464242499731736163120, −6.44999684693650898729159137260, −5.66724962518604070326983010603, −5.02233713502525619373888536732, −3.28520776811565629481700952265, −2.58569412724386643584708167913, −1.68905629961158643117151180384, 0.75446338838500164473618033274, 2.531136270305897227713273410726, 3.74042440036715086270589183913, 4.54696607729419709352450637787, 5.14108448375517956074734816075, 6.0226424307480245964909809024, 7.32161793659324244231348878143, 8.164052916203969340281252038224, 9.62988175025557310627105439379, 10.235074527651924855960440060661, 11.46887528920675395940923099768, 11.885869214646404254157781228975, 13.06368127869969332120612995073, 13.821451792300758170541035395870, 14.6074743033305779986281789445, 15.82752435957271802565564356781, 16.24521466176343523292078353210, 16.85586780043345774045173929577, 17.734077920029265342606768740931, 19.37517722544162405948371299133, 20.379966130269428424705068822655, 20.686210079288669280871875152452, 21.4880618855147452323569155700, 22.31391395640895208833998602872, 23.1564381575442402034980130811

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