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Properties

Label 1-571-571.23-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.655 + 0.755i$
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  + (0.245 + 0.969i)2-s + (−0.909 + 0.416i)3-s + (−0.879 + 0.475i)4-s + (0.701 + 0.712i)5-s + (−0.627 − 0.778i)6-s + (−0.340 − 0.940i)7-s + (−0.677 − 0.735i)8-s + (0.652 − 0.757i)9-s + (−0.518 + 0.854i)10-s + (−0.518 − 0.854i)11-s + (0.601 − 0.799i)12-s + (0.863 − 0.504i)13-s + (0.828 − 0.560i)14-s + (−0.934 − 0.355i)15-s + (0.546 − 0.837i)16-s + (0.0495 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.245 + 0.969i)2-s + (−0.909 + 0.416i)3-s + (−0.879 + 0.475i)4-s + (0.701 + 0.712i)5-s + (−0.627 − 0.778i)6-s + (−0.340 − 0.940i)7-s + (−0.677 − 0.735i)8-s + (0.652 − 0.757i)9-s + (−0.518 + 0.854i)10-s + (−0.518 − 0.854i)11-s + (0.601 − 0.799i)12-s + (0.863 − 0.504i)13-s + (0.828 − 0.560i)14-s + (−0.934 − 0.355i)15-s + (0.546 − 0.837i)16-s + (0.0495 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.655 + 0.755i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.655 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4380490056 + 0.9605760080i\)
\(L(\frac12)\) \(\approx\) \(0.4380490056 + 0.9605760080i\)
\(L(1)\) \(\approx\) \(0.6962415725 + 0.5845763817i\)
\(L(1)\) \(\approx\) \(0.6962415725 + 0.5845763817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.245 + 0.969i)T \)
3 \( 1 + (-0.909 + 0.416i)T \)
5 \( 1 + (0.701 + 0.712i)T \)
7 \( 1 + (-0.340 - 0.940i)T \)
11 \( 1 + (-0.518 - 0.854i)T \)
13 \( 1 + (0.863 - 0.504i)T \)
17 \( 1 + (0.0495 + 0.998i)T \)
19 \( 1 + (0.115 + 0.993i)T \)
23 \( 1 + (0.490 + 0.871i)T \)
29 \( 1 + (-0.401 + 0.915i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (-0.213 + 0.976i)T \)
41 \( 1 + (-0.986 - 0.164i)T \)
43 \( 1 + (0.922 - 0.386i)T \)
47 \( 1 + (0.546 - 0.837i)T \)
53 \( 1 + (0.371 + 0.928i)T \)
59 \( 1 + (0.789 - 0.614i)T \)
61 \( 1 + (0.997 + 0.0660i)T \)
67 \( 1 + (0.371 + 0.928i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.213 + 0.976i)T \)
79 \( 1 + (-0.724 + 0.689i)T \)
83 \( 1 + (-0.627 - 0.778i)T \)
89 \( 1 + (-0.627 - 0.778i)T \)
97 \( 1 + (0.991 + 0.131i)T \)
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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

−22.74480561117240690994406244309, −22.1775920230273209231428328036, −21.14230622840118533106457503317, −20.81338095915255251273157762484, −19.57247139232172660236787970636, −18.66705969460143267227186765539, −18.038105651286652168665491342463, −17.45918362593320593233921043479, −16.26386315406353939683668671605, −15.48649439849238521861127811815, −14.063207950520113243058247952504, −13.19036471688372637272467705015, −12.687399014775164675141177601974, −11.91498115577573619616565483551, −11.13752290889049868949279105627, −10.0746365425843886282509888708, −9.316965995472865726267263685753, −8.48142919310161476437892638086, −6.87749061284278922980709754102, −5.863531845199553188356836765498, −5.11777059109507809509522743831, −4.436988356888025194151809849397, −2.65764370188565941660889042988, −1.92890482727146931822754142232, −0.7033443067332565109435970085, 1.09873248656919047607391677038, 3.361702957290672875070305502957, 3.86503834072710314659611147626, 5.35977211963181397790960047661, 5.89602625822625749389909100210, 6.64318284655248647773280373898, 7.569806178238511404481861871513, 8.72423543167403348521202073620, 10.02524034098965173830691518745, 10.43306355373758150747197880445, 11.43124167543511870538846302253, 12.88578858185612586523176231735, 13.39654642929020902559260597056, 14.30055245175956449730128748043, 15.30939416569483883036846761096, 15.998276238439799819200939255382, 17.03417321759354134169560625388, 17.22902595399650941234415648606, 18.40707429648599705679047762423, 18.85581477841872112100115098162, 20.62381150910050572359179671229, 21.371961808322608588849604004558, 22.13275779521066345747597050935, 22.805974271285191880321674674341, 23.50937654423083366853533258804

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